Abstract

Title of Dissertation: Interface Motion Driven by Curvature and Diffusion: Analytic Dependence on the Initial Data for the Mullins-Sekerka Equation

Sergio A. Alvarez, Doctor of Philosophy, 1996

Dissertation directed by: Professor Robert L. Pego Interdisciplinary Applied Mathematics Program

The Mullins–Sekerka problem is a fully nonlinear, nonlocal free–boundary problem which arises in the study of phase transitions in materials science, as a model of the motion of the phase boundaries or interfaces which separate regions of constant phase during, for example, solidification (W.W. Mullins, R.F. Sekerka. “Morphological stability of a particle growing by diffusion or heat flow”, Journal of Applied Physics, vol 34, 1963, 323–329). The problem involves the motion of a closed surface inside a bounded domain, driven by its mean curvature and by instantaneous diffusion. We prove local existence, uniqueness, and analytic dependence on the initial data of solutions of the Mullins–Sekerka problem in the case of spatial dimension two, for initial curves which are small perturbations of a disjoint union of circles. The method of maximal regularity (see A. Lunardi, “Analytic Semigroups and Optimal Regularity in Parabolic Problems”, Birkhäuser, 1995; S. Angenent, “Nonlinear Analytic Semiflows”, Proceedings of the Royal Society of Edinburgh, vol 115A, 1990, 91–107) is used in Banach spaces of periodic functions with algebraically decaying Fourier transforms. The analytic core of the proof is the analysis of the integral operators of potential theory in these nonstandard function spaces. The thesis also contains an analysis of the Dirichlet-to-Neumann mapping for multiply connected planar domains in this context. Various results concerning the spaces used in the thesis are given, concerning products, quotients, and analytic images of the associated functions, as well as interpolation properties, and compactness and smoothness of integral operators on these spaces.

Interface Motion Driven by Curvature

and Diffusion: Analytic Dependence

on the Initial Data for the

Mullins-Sekerka

Equation

by

Sergio A. Alvarez

Dissertation submitted to the Faculty of the Graduate School of the

University of Maryland at College Park in partial fulfillment

of the requirements for the degree of Doctor of Philosophy 1996

Advisory Committee:

Professor Robert L. Pego, Chairman/Advisor Professor Stuart S. Antman Professor Manoussos N. Grillakis Professor Ricardo H. Nochetto Professor James A. Reggia

\begin{matrix}c \\ ◯ \end{matrix} Copyright by Sergio A. Alvarez

Dedication

To Carolina, who gives my world meaning

Acknowledgements

I wish to thank Robert Pego for having provided such expert direction during this journey. He led me from my initial vague desire “to study evolutionary partial differential equations as if they were ordinary differential equations”, to a well–chosen mathematical problem, complete with the seeds of a map of the terrain. He had the insight that the method of maximal regularity might lead to smooth dependence on parameters for solutions of the Mullins–Sekerka problem. Once I had started, he helped me see the road more clearly, often letting me find the way even if he knew exactly where to go. He was always available for discussion and constantly suggested new ideas. This was crucial during some of the more difficult times. He has brought me closer to the great rivers of mathematical analysis, and for this too I am grateful. Finally, I acknowledge his financial support which allowed me to concentrate on the research for this dissertation.

I thank the other members of my dissertation committee, Stuart Antman, Manoussos Grillakis, Ricardo Nochetto, and Jim Reggia, for their time and their comments. Thanks are also due Mike Fitzpatrick, who would have served on the committee as well, had it not been for an unfortunate timing conflict.

I thank Jaime Lesmes and my other friends at the Universidad de los Andes for giving me my start in mathematics, and Michael Crandall and Constantin Dafermos for inspiring me to work in partial differential equations.

I thank Peter Bates for helpful information he provided during the 1995 IAS Park City Mathematics Institute. Jeff Dodd and Katie Gurski sat through test runs of seminar talks related to my thesis work, and I thank them for their useful comments.

In the Computer Science Department at Maryland, my second home on the College Park campus, I wish to thank Jim Reggia and his group for creating an exciting environment which not only allowed me to become involved in some very interesting computer science research, but also greatly contributed to the successful completion of this dissertation. I would also like to offer special thanks to Jack Minker and his students for the enormous support that they provided, in many forms, for my wife and for me throughout our stay at College Park.

Many other people contributed to making my years at Maryland varied and interesting. I thank Bill Smyth, Ram Nair and his family, Cecilia Luna, Luz Marina Sorensen, and Jane Henry for their friendship, and John Horváth, Ken López–Escobar, and John Maddocks for making things easier and more pleasant in one way or another. I thank the staff of the Mathematics and Computer Science Departments at the University of Maryland at College Park for cheerfully dealing with enormous amounts of paperwork on my behalf.

I cannot imagine going through life without the love of my wife Carolina. She shared my experiences as we worked toward our doctorates, and has enriched my life in many other ways. It has given me great joy to see her blossom into a researcher while her many other bright lights continue to shine undiminished. To her I offer my profound gratitude for the past and present, and a toast to the future. My parents and hers made it possible for us to pursue our studies with their constant spiritual and financial support. I am forever indebted to them.

I acknowledge financial support from The Graduate School at the University of Maryland, College Park, which awarded me a Graduate School Fellowship for my first two years of doctoral study, and from the Mathematics Department at Maryland, which gave me a supplementary fellowship for the first three.

Table of Contents

1 Overview: well–posedness for the Mullins–Sekerka interfacial motion model by the method of maximal regularity 1 1.1 The Mullins–Sekerka problem . . . . . . . . . . . . . . . . . . . . 2 1.2 Contribution of the thesis . . . . . . . . . . . . . . . . . . . . . . 4 1.3 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.4 Sketch of the proof . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.4.1 Maximal regularity method . . . . . . . . . . . . . . . . . 10 1.4.2 Main technical steps . . . . . . . . . . . . . . . . . . . . . 14

2 The spaces E^{h} of functions with algebraic Fourier decay 19 2.1 Motivation: failure of maximal regularity in L^{p} . . . . . . . . . . . 20 2.2 Definition and basic properties of the

spaces E^{h} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.3 Analyticity in E^{h} of multiplication and of composition with ana-

lytic functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.4 The O(θ^{2}) lemma and E^{h} analyticity of division by vanishing

denominators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.5 Integral operators with E^{h} kernels . . . . . . . . . . . . . . . . . . 45 3 Analyticity of the Dirichlet to Neumann map and maximal

regularity for the Mullins–Sekerka equation 50 3.1 Problem formulation . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2 The case of a single loop . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Analyticity of the Dirichlet to Neumann map . . . . . . . 60 3.2.2 Maximal regularity for the Mullins–Sekerka vector field . . 70 3.2.3 The case of a bounded domain . . . . . . . . . . . . . . . . 75

3.3 The case of multiple loops . . . . . . . . . . . . . . . . . . . . . . 89 3.3.1 Analyticity of the vector field . . . . . . . . . . . . . . . . 91 3.3.2 Maximal regularity . . . . . . . . . . . . . . . . . . . . . . 98

Bibliography 104 Interface Motion Driven by Curvature and Diffusion: Analytic Dependence

on the Initial Data for the Mullins-Sekerka Equation

Sergio A. Alvarez

August 5, 1996

This comment page is not part of the dissertation.

Typeset by L^{A}TEX using the dissertation class by Pablo A. Straub, University of

Maryland.

Chapter 1

Overview: well–posedness for the

Mullins–Sekerka interfacial motion

model by the method of maximal

regularity

In the study of phase transitions, the issue arises of describing the motion of the regions of transition which separate the different phases, such as that between ice and liquid water. We are concerned with a specific model of the dynamics of the transition region, commonly known as the Mullins–Sekerka model [25]. This is a sharp interface model. In other words, there is assumed to be a surface (a curve in the two–dimensional case), called the interface, across which the phase changes abruptly, and which separates space into two regions in each of which the phase is constant. Mathematically, the Mullins–Sekerka model leads to a fully nonlinear, nonlocal free boundary problem for a scalar function representing the temperature field. It is a quasistatic version of the well–known Stefan problem.

In the present work we prove for the first time that solutions of the Mullins– Sekerka problem depend \underline{analytically} on the initial curve, in the case of initial curves which are close, in a sense defined below, to the disjoint union of a collection of circles. Furthermore. our existence and uniqueness proofs are considerably simpler than those obtained recently by Chen et al. [9], [10]. We provide a conceptually transparent mapping–theoretic argument based on explicitly linearizing the Mullins–Sekerka vector field. Our work also contains a new analysis of the Dirichlet-to-Neumann mapping for planar domains.

In this introductory chapter, we define the Mullins–Sekerka free-boundary problem and discuss the contributions of the thesis, then briefly describe some physical background. Finally, we provide a sketch of the proof of the main theorem, including a description of the maximal regularity method which we will use to prove existence, uniqueness, and analytic dependence on parameters for solutions of the Mullins–Sekerka problem.

1.1 The Mullins–Sekerka problem

The two–dimensional Mullins–Sekerka interface motion problem is defined in the following way. At each instant of time t, the interface is assumed to be a closed curve Γ(t) having one or more connected components, each diffeomorphic to a circle, lying within a given simply connected bounded domain Ω in the Euclidean plane. The curve undergoes temporal evolution driven by its curvature function κ, as follows. The curvature function is taken as the Dirichlet datum on the curve for the Laplace equation inside the surrounding domain, with an auxiliary Neumann condition on the boundary. Thus, one needs to solve the boundaryvalue problem

Δu = 0 in Ω \ Γ
u = κ on Γ
\frac{∂u}{∂n} = 0 on ∂Ω

(1.1)

The normal velocity of the curve at each point is then required to equal to the jump across the curve of the normal derivative of the resulting harmonic function:

V_{n} = [\frac{∂u}{∂n}]_{Γ}  (1.2)

In equation 1.2, the jump is defined as

[\frac{∂u}{∂n}]_{Γ} = \frac{∂u}{∂n}|_{Γ^{+}} − \frac{∂u}{∂n}|_{Γ^{−}}  (1.3)

with n denoting the outward pointing unit normal to Γ.

The above Mullins–Sekerka prescription for interface motion is similar to the motion by mean curvature, defined by:

V_{n} = κ on Γ  (1.4)

However, in the Mullins–Sekerka motion, the curvature does not locally determine the velocity of the interface; rather, a diffusion process mediates between the curvature and the resulting velocity. The velocity is determined by the asymptotic state reached after the diffusion has been completed. This non– locality makes the analysis of well–posedness for the Mullins–Sekerka problem more difficult than for motion by mean curvature.

1.2 Contribution of the thesis

In this thesis we prove existence, uniqueness, and analytic dependence on parameters for classical solutions of the Mullins–Sekerka initial-boundary-value problem described above, assuming small initial data.

The main result

The main result of the thesis is the following. The phase spaces E^{h} are introduced below in Definition 2.2.1; roughly, curves in E^{h} are h times differentiable.

Theorem 1.2.1. Assume that α > 4. Let Ω be a bounded starshaped domain in the Euclidean plane with an E^{α} boundary. There exists r > 0 such that if Γ(0) is a closed E^{α} curve inside Ω, whose E^{α} distance to the disjoint union of some collection of n circles is less than r, then there exists T > 0 for which there is a unique continuous family Γ : [0, T] → E^{α}(⊔_{n}S^{1}, ℝ^{2}) of planar closed E^{α} curves satisfying the Mullins–Sekerka problem defined above for times t ∈ [0, T]. Furthermore, the mapping Γ(0) ↦ (Γ(t))_{t∈[0,T]} is analytic as a mapping from E^{α} to C([0, T], E^{α}).

Embedding properties of the spaces E^{h} yield a slightly modified statement in Hölder spaces. Namely, assuming that the initial curve Γ(0) is in C^{α}(⊔_{n}S^{1}, ℝ^{2}), one obtains a unique solution in C([0, T], C^{α−1−ϵ}(⊔_{n}S^{1}, ℝ^{2})), which depends analytically on Γ(0) in the associated Banach structures. Above, each connected curve is assumed to be parametrized in polar coordinates (r, θ). Even for curves with multiple components, we let r ∈ ℝ^{n} denote the vector of radial distances relative to preselected points, one inside each connected component, while θ is a scalar which denotes the polar angle. The interface is parametrized at each fixed time in the form:

r = r_{0} + d(θ),  (1.5)

for some constant vector r_{0} ∈ ℝ^{m} and some function d : S^{1} → ℝ^{m} which belongs to the space E^{a} for some a > 4. See chapter 3 for more information about the parametrization. The choice of parametrization is important because, as shown in chapter 2, the phase spaces E^{h} are not invariant under C^{∞} changes of variables.

This thesis also provides a new analysis of the Dirichlet to Neumann mapping for planar domains. We prove the following result, which should have applications in contexts other than that of the present work. N(d) denotes the mapping from the Dirichlet datum κ to the jump of the corresponding Neumann datum [∂u/∂n]_{Γ} in Eqs. 1.1, 1.3, where the interface curve Γ has the polar coordinate parametrization r = r_{0} + d(θ) given above in Eq. 1.5. This mapping is discussed in chapter 3. Theorem 1.2.2. There exists R ∈ ℝ^{+} such that if d ∈ E^{a} with a > 4 and ||d||_{E^{a}} < R, then the Dirichlet-to-Neumann map N(d) maps E^{a−2} continuously into E^{a−3}. N(d) is analytic as a map d ↦ N(d) from the ball of radius R in E^{a} to C(E^{a−2}, E^{a−3}).

1.3 Background

In this section we describe some of the work that has previously been done on the Mullins–Sekerka problem. Work connecting the Mullins–Sekerka problem to other interface models is discussed also in order to give the reader a glimpse of the physical context.

Related work

Some well–posedness issues for the Mullins–Sekerka problem have been addressed recently by other authors. Existence of weak solutions was obtained by Chen [9], assuming a single component for the interface curve and no exterior boundary. Chen, Hong, and Yi [10] have proved existence and uniqueness of classical solutions in Euclidean spaces of any dimension. The issue of parameter dependence has remained open and has only now been addressed with the work contained in this thesis. Other works related to the Mullins–Sekerka problem include that of Constantin and Pugh [11], who showed small–data existence of solutions globally in time for the related Hele–Shaw problem in the case of a connected analytic curve close to a circle. Mayer [21] showed that convexity is not preserved by the single phase Mullins–Sekerka flow. Bates, Chen, and Deng [6] developed a semi–implicit numerical scheme for the Mullins–Sekerka problem in the case of the plane ℝ^{2}.

Relation to diffuse interface models

Certain sharp interface models, including the Mullins–Sekerka model, are asymptotic limits of diffuse interface models, in which a gradual transition is allowed between phases. In brief, the diffuse interface models we are referring to arise as follows; see Penrose and Fife [28], and, for a detailed discussion of models of interface motion in the two–dimensional case from the point of view of rational continuum thermodynamics, see Gurtin [16].

One begins with a free-energy functional of the form

F (ϕ) = ∫_{Ω} dx (f(ϕ(x)) + \frac{1}{2}|∇ϕ(x)|^{2}) ,  (1.6)

where ϕ is the phase field. Typically, the values ϕ = ±1 correspond to the two distinct phases of the sharp interface models; here, the phase varies continuously, and the interface corresponds roughly to the level set ϕ = 0. Then it is postulated that the phase evolution is given by the associated gradient flow

\frac{∂ϕ}{∂t} = −δ_{ϕ}F (ϕ) = Δϕ − f′(ϕ).  (1.7)

For example, in the case of a standard double–well potential

f(ϕ) = \frac{1}{4}(ϕ^{2} − 1)^{2},  (1.8)

this yields the dynamical equation

\frac{∂ϕ}{∂t} = Δϕ − ϕ(ϕ^{2} − 1),  (1.9)

also known as the Allen–Cahn equation. If the variables in eq. 1.9 are rescaled: t ↦ ϵ^{2}t, x ↦ ϵx, then in the limit as ϵ → 0, the interface moves according to its mean curvature: if V denotes the normal velocity of the interface at each point, and if κ denotes the mean curvature of the interface, one has:

V = κ  (1.10)

This convergence has been studied by several authors, including de Mottoni and Schatzman [23], [24]. Evans, Soner, and Souganidis [14] proved that the asymptotic motion of the level sets of the Allen–Cahn equation is given by the above mean curvature equation in a generalized sense, even after singularities have developed. Also see Ilmanen [17] for a generalization to varifolds.

A related equation results if instead of gradient flow, one postulates that the phase evolution satisfies

\frac{∂ϕ}{∂t} = −Δδ_{ϕ}F (ϕ) = Δ (f′(ϕ) − Δϕ) .  (1.11)

This evolution law guarantees that the spatial integral

∫_{Ω} ϕ(x) dx  (1.12)

is conserved, if appropriate boundary conditions are enforced. Choosing f to be the double–well potential in eq. 1.8 then leads to the Cahn–Hilliard model of phase separation:

\frac{∂ϕ}{∂t} = Δ (ϕ(ϕ^{2} − 1) − Δϕ) .  (1.13)

There have been several works showing that the Cahn–Hilliard equation leads to the Mullins–Sekerka problem defined below in the sharp interface limit. Pego [27] showed that the Mullins–Sekerka problem arises formally when analyzing the following version of the Cahn–Hilliard equation

\frac{∂ϕ}{∂t} = Δ (f′(ϕ) − ϵ^{2}Δϕ)

on a slow time scale using matched asymptotic expansions in the parameter ϵ. Stoth [32] proved convergence as ϵ → 0 of the Cahn–Hilliard dynamic to the Mullins–Sekerka problem in the special case of radial symmetry. Alikakos, Bates, and Chen [2] proved convergence in the general case on any time interval on which classical solutions of the Mullins–Sekerka equation exist.

Models of Mullins–Sekerka type arise as sharp interface limits of the more complicated phase–field models as well, as shown formally by Caginalp [8] and rigorously by Soner [30].

1.4 Sketch of the proof

We now provide a skeleton of the proof of well–posedness. Detailed proofs of the statements made here are given in the following chapters. We consider curves which are small perturbations (in a sense made precise below) of the disjoint union of a collection of circles. For the sake of expository simplicity, we will assume here that the interface curve is connected and differs only slightly from a single circle of radius r_{0}. We also ignore the auxiliary Neumann condition on the boundary. The general case presents some additional difficulties; it is treated in detail in chapter 3.

We parametrize the curve by giving the polar radius r as a function of the polar angle θ:

r = r_{0} + d(θ)

where r denotes the radial distance from the center of the reference circle, and θ varies between, say, −π and π. Thus, the curve is described by a mapping from the circle S^{1} to the plane ℝ^{2}. Letting N(d) denote the Dirichlet-to-Neumann mapping taking Dirichlet to Neumann data for the domain enclosed by the curve parametrized by the function d, and letting κ(d) denote the curvature function of the curve, we may write the Mullins–Sekerka equation as

∂_{t}d = \frac{\sqrt{(r_{0} + d)^{2} + d′^{2}}}{(r_{0} + d)} [N(d)κ(d)]

The brackets [ ] denote the jump across the curve of the enclosed quantities. Although there are really two different Dirichlet-to-Neumann mappings here, one for the interior domain, and one for the exterior, we circumvent the need to consider these mappings separately by making use of the jump conditions for single layer potentials (see chapter 3). Accordingly, we will drop the brackets [ ] for the jump, and use the notation N(d) to refer to the jump of the usual Dirichlet-to-Neumann map.

The well–posedness issue for the Mullins–Sekerka problem is difficult due to the fully nonlinear curvature term as well as the nonlocality and nonlinearity in d of the Dirichlet-to-Neumann mapping. The curvature operator is given in polar coordinates by

κ(d) = \frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}}  (1.14)

The two-phase Dirichlet-to-Neumann mapping N(d) is a nonlocal operator which we will describe in chapter 3 using single layer potential theory.

1.4.1 Maximal regularity method

We address the well–posedness problem by the method of maximal regularity (see Lunardi [20] and Angenent [3]). For another recent example of the application of this method to obtain local well–posedness for an interface motion problem, see [4].

We now give a brief description of the maximal regularity method. Assume given a (nonlinear) vector field F⃗ which is defined on a Banach space Y continuously and densely embedded in a Banach space X. The pair (X, Y) will be referred to as the phase pair for the dynamical system associated with F⃗. We are interested in studying the existence, uniqueness, and parameter dependence for solutions of the initial value problem

\frac{du}{dt} = F⃗ (u(t))  (1.15)
u(0) = u_{0}  (1.16)

for initial conditions u_{0} lying in a neighborhood in X of the origin 0.

The maximal regularity method is based on using the variation-of-constants formula

u(t) = ∫_{0}^{t} e^{(t−s)L}f(s)ds  (1.17)

for the solution of the forced, or nonhomogeneous, initial value problem

∂_{t}u = Lu + f, u(0) = 0

associated with the linearization L := dF⃗ (0) of the vector field F⃗ at the origin 0. The maximal regularity method requires several properties to be satisfied by the nonlinear vector field and its linearization.

Differentiability assumption. It is assumed that the nonlinear vector field F⃗, i.e. in the case of the Mullins–Sekerka problem ^{\frac{\sqrt{(r_{0}+d)^{2}+d′^{2}}}{(r_{0}+d)}} N(d)κ(d), is C^{k} as a mapping from Y to X, for some k ≥ 1.

Abstract parabolicity assumption. The forward time flow generated by the linearization L should define an analytic semigroup on X. The latter requirement is equivalent to a spectral property of L; namely, L should be sectorial, in the sense that the spectrum of L is contained in a wedge in the complex plane with vertex on the real line and opening toward the left half plane with an angle less than π radians [26]. The main maximal regularity requirement is the following one, which in the sequel we will refer to simply as the MR property.

MR property. The linear operator L is said to satisfy the MR property if and only if the integral on the right hand side of the variation-of-constants formula eq. 1.17 defines a bounded linear operator from the space C(I, X) of continuous X–valued curves into the space C(I, Y) of continuous Y –valued curves, for some real interval I = [0, T]. Under the above assumptions, the maximal regularity method guarantees by a contraction mapping argument that well–posedness holds for the initial value problem eq. 1.16 for the original nonlinear equation, for initial data near the linearization base point. Furthermore, the mapping from initial data to solution trajectories in phase space is shown to be as smooth as the nonlinear vector field, in other words, it is C^{k}. The smoothness index k can be a natural number, ∞, or even ω, i.e. analytic.

The challenge in applying the method is to find appropriate spaces X,Y and to prove that the maximal regularity property holds in these spaces. The small data restriction arises because it is necessary to prove that the MR property holds near each desired initial datum. This requires detailed information about the linearized field which is difficult to obtain except at special solutions.

Failure of maximal regularity in reflexive phase spaces. Heuristically, in the case of the Mullins–Sekerka equation the maximal regularity estimate may be obtained by choosing Y to be a space of functions which have 3 orders of smoothness more than the members of X, as the following sketch suggests. One shows that the linearization of the Mullins-Sekerka equation around a single circle is of the form

∂_{t}d = Ld

where, in terms of the Fourier transform ˆ( ), the linear operator L satisfies

L̂f_{k} = \frac{2π}{r_{0}^{3}} |k|(1 − k^{2})f̂_{k}

For large |k|, the factor on the right hand side is approximately −|k|^{3}. Thus, the k-th Fourier coefficient of the variation of constants integral is approximately bounded by

∫_{0}^{t} e^{−|k|^{3}(t−s)}|f̂_{k}(s)|

which after integration should yield a result of order |k|^{−3}|f̂_{k}|, corresponding to a function which has three orders of smoothness more than f.

Nonetheless, this heuristic argument is not correct for some very natural choices for the spaces X,Y; for instance, we show in the following chapter that the choice X = L^{2}, Y = W^{3,2} fails. In fact, a theorem by Baillon [5] implies that no reflexive space can satisfy the maximal regularity property, independently of the dynamical equation being considered. Baillon’s theorem states that if there exists an unbounded operator on the phase space X which satisfies the MR property, then X must contain an embedded copy of the sequence space c_{0}; see [13] for a simplified proof of this fact. A successful choice of spaces for the Mullins–Sekerka problem is described below.

It is important to note that if in the MR property one requires boundedness of the variation of constants integral 1.17 on spaces of curves such as L^{p}(I, X), instead of C(I, X), then it becomes possible to choose reflexive spaces X as phase spaces without violating the resulting modified MR property; see e.g. [12]. Throughout the present work we will be concerned with the case of spaces of continuous curves as in the definition of the MR property above.

Perturbation stability of maximal regularity. To prove that MR holds for a given operator L, it is often convenient to write L = T + S, where T is known to satisfy MR and S is a “smooth” perturbation. In fact, we will use this technique in chapter 3 to extend the basic maximal regularity argument to include the effect of a Neumann condition on the exterior boundary, and of multiple interacting component loops of the interface curve. The basic MR perturbation result which we will use is the following.

Theorem 1.4.1. (Angenent [3]). If T satisfies MR relative to the phase pair (X, Y), and if S : Y → X is such that for every ϵ > 0 there exists C_{ϵ} ∈ ℝ^{+} such that

||Sy||_{X} ≤ ϵ||y||_{Y} + C_{ϵ}||y||_{X} ∀y ∈ Y,

then T + S also satisfies MR relative to (X, Y).

As observed in [3], the property in theorem 1.4.1 follows immediately if one can show that the operator S is continuous from some interpolation space between X and Y strictly below Y, into X.

1.4.2 Main technical steps

To begin with, one needs to choose the Banach spaces X,Y to be used as a phase pair for the Mullins–Sekerka dynamical system. As we show at the beginning of the next chapter, the traditional L^{p} Sobolev spaces are not suitable. Instead, we use spaces based on those used by Bricmont, Kupiainen, Lin in [7]. Namely, we consider the spaces E^{h} of functions f defined on the circle S^{1}, whose Fourier transform f̂_{k} is of order o(|k|^{−h}) as |k| → ∞, normed by the c_{0} norm on the Fourier transform side. We prove several useful compactness, embedding and interpolation properties of these spaces in the following chapter.

Proving maximal regularity for the Mullins–Sekerka vector field involves proving two statements:

• Differentiability: The nonlinear vector field F⃗ (d) = ^{\frac{\sqrt{(r_{0}+d)^{2}+d′^{2}}}{(r_{0}+d)}} N(d)κ(d) maps E^{h} to E^{h−3} analytically in d.

• Abstract parabolicity and maximal regularity: The linearized vector field L = δ (^{\frac{\sqrt{(r_{0}+d)^{2}+d′^{2}}}{(r_{0}+d)}} N(d)κ(d))_{d=0} defines a sectorial operator and satisfies the MR property defined in 1.4.1.

Differentiability of the nonlinear vector field

To address the first issue, we treat the curvature term and the Dirichlet to Neumann map separately.

Curvature term. The nonlinearities in the curvature term κ(d) involve products and quotients by nonvanishing denominators. The first are handled by showing

Lemma. (Theorem 2.3.1). Let h ≥ 0. The product fg is in E^{h} whenever both factors f,g are in E^{h}, if at least one of the factors, say g, is in E^{μ} for some μ > 1. The mapping (f, g) ↦ fg is analytic as a mapping E^{h} × E^{μ} → E^{h}.

The second type is handled by applying the following result with Φ given by inversion.

Lemma. (Theorem 2.3.3). Let h > 2. If Φ is a mapping which is analytic on some open interval S in ℝ, then the composition Φ∘f is in E^{h} whenever f ∈ E^{h} and the range of f is contained in the set S. The mapping f ↦ Φ ∘ f is analytic from the open subset of E^{h} consisting of functions whose range is contained in S to E^{h} itself.

These lemmas imply that the curvature term κ(d) is analytic as a map from a neighborhood of the origin in E^{h} to E^{h−2}. See section 3.1.

Dirichlet-to-Neumann map. In the treatment of the single and double layer integrals which arise when studying the Dirichlet–to–Neumann map using potential theory, one needs to deal with quotients of the form O(θ^{2})/O(θ^{2}). Our main result in this direction in the spaces E^{h} is:

Lemma. (Theorem 2.4.1). Let h > 3. If f ∈ E^{h} and g ∈ E^{h}, with both f(θ) and g(θ) of order O(θ^{2}), and 1/g of order 1/θ^{2} as θ → 0, then the quotient f/g is in E^{h−2}. The mapping (f, g) ↦ f/g is analytic between the naturally associated infinite–dimensional manifolds.

Using this lemma, we infer that the Dirichlet to Neumann map is analytic as a map from a neighborhood of the origin in E^{h} to the space of mappings C(E^{h−2}, E^{h−3}), for h > 4. The restriction on h arises because the result 2.3.3 on analyticity of composition requires that we work in E^{a} with a > 2, and in our analysis of the Dirichlet-to-Neumann map in chapter 3 we must apply this result with a = h − 2.

Combining our results on the operators κ and N, we see that the Mullins– Sekerka vector field is analytic as a map from an E^{h} neighborhood of 0 to E^{h−3} if h > 4. This proves the first of the two statements making up the proof of well–posedness.

Abstract parabolicity and maximal regularity for the linearized vector field

Next we address the maximal regularity issue. The spaces E^{h} are extremely well suited to this task. Examining the variation of constants formula, it becomes clear that in order to prove maximal regularity with respect to the phase pair (E^{h−3}, E^{h}), it suffices to show that the linearization L of the Mullins–Sekerka vector field has a Fourier symbol L̂_{k} which is of order |k|^{3}|d̂_{k}| uniformly in k. The sufficiency of this condition is obvious from the definition of the spaces E^{h}. The condition is \underline{not} sufficient in L^{p} Sobolev spaces, for example (see the following chapter).

Since for a circle (d = 0) the curvature κ is constant, and since the solution of the Dirichlet problem with constant boundary datum is constant, one has N(d)κ(0) = 0. It follows that the linearized vector field L at d = 0 reduces to

δF⃗ (d)_{d=0} = δ (N(d)κ(d))_{d=0} = N(0)κ′(0) = −\frac{2π}{r_{0}}DHκ′(0)

Here, D denotes differentiation with respect to the polar angle θ. H denotes the Hilbert transform, a convolution operator with the Fourier symbol −\frac{i}{2}sgnk. The linearized curvature operator κ′(0) is simple enough (see section 3.1):

κ(d) = \frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}} ⟹ κ′(0) = \frac{−d′′ − d}{r_{0}^{2}}

This shows that L is a sectorial operator and therefore generates an analytic semiflow, as required by the abstract parabolicity assumption. Also, the claim concerning the asymptotic behavior for large |k| of the symbol of L now follows easily, and the maximal regularity statement holds.

Dealing with the zero flux boundary condition and curves with multiple components

We show in chapter 3 that the net effect of the Neumann condition on the exterior boundary of the domain surrounding the interface curve, and of the disconnectedness of the curve, is to introduce certain operators which are smoothing in the sense that they are continuous on an E^{h} interpolation space as explained in the comments following Theorem 1.4.1. The argument makes use of an interpolation result for the E^{h} spaces (Theorem 2.2.2) which we prove in chapter 2. This allows us to conclude that the linearized Mullins–Sekerka vector field satisfies the abstract parabolicity and MR properties in the general case.

The spaces E^{h} of functions with

algebraic Fourier decay

We begin this chapter with a proof that L^{p} Sobolev spaces cannot yield maximal regularity, even for the simplest parabolic partial differential equation. With this fact as motivation, we introduce the E^{h} spaces which we will use as phase spaces for the Mullins–Sekerka flow, and give some of their basic properties, such as inclusion and interpolation relations, the analyticity of multiplication as a map E^{a} × E^{b} → E^{min(a,b)}, and analyticity of division by vanishing denominators of size O(x^{2m}) as a map O_{a}(x^{2m}) × Θ_{a}(x^{2m}) → E^{a−2m} (these spaces are defined below; Θ(x^{2m}) just means that the functions are O(x^{2m}) but not smaller than that). At the end of the chapter, some results on integral operators are given, showing that smoothness properties of the kernel carry over to the corresponding operator between the appropriate E^{h} spaces; the compactness of certain E^{h} integral operators is also proven. The various properties of the E^{h} spaces proven here will be used in the potential–theoretic study of the Mullins–Sekerka vector field in the following chapter.

2.1 Motivation: failure of maximal regularity

in L^{p}

This section contains an example by Robert L. Pego and Sergio A. Alvarez, based on an idea of Vladimir Sverák. We refer to [3] and [20] for the theory of maximal regularity. The basic definitions are given above in the introduction. For the present purposes, it suffices to recall that a continuous linear vector field L : E_{1} → E_{0}, where E_{1}, E_{0} are Banach spaces with E_{1} densely contained in E_{0}, satisfies the maximal regularity property (MR), iff L generates an analytic forward–time flow on E_{0} and the rule

T : f ↦ ∫_{0}^{t} e^{(t−s)L}f(s)ds

maps C(I, E_{0}) continuously to C(I, E_{1}) for some interval I ⊆ ℝ. The interest of this property in the context of nonlinear partial differential equations is that local existence, uniqueness and smooth dependence on parameters can easily be obtained for operators whose linear part satisfies MR.

A theorem by Baillon [5] states that if a vector field L satisfying MR exists as above, with L unbounded as an operator on the phase space E_{0}, then E_{0} must contain an embedded copy of the sequence space c_{0}. Here, we carry out the exercise of showing how MR fails for the 1–D Laplacian with E_{0} = L^{p}(ℝ) and E_{1} = the Sobolev space W_{p}^{2}(ℝ) where 1 ≤ p ≤ 2.

First consider the following sketch of a “proof” that the Laplace operator ∂_{x}^{2} actually does satisfy MR with respect to (W_{2}^{2}, L^{2}). Start with f ∈ C(I, L^{2}). In terms of the Fourier transform ˆ( ), the map T defined above by convolution with the fundamental solution operator e^{t∂_{x}^{2}} satisfies

T̂ f(t, ξ) = ∫_{0}^{t} e^{−ξ^{2}(t−s)}f̂(s, ξ) ds

The W_{2}^{2} norm of T f(t, ·) is controlled by the L^{2} norm of ∂_{x}^{2}T f(t), whose square equals

∫_{ℝ} dξ |∫_{0}^{t} ξ^{2}e^{−ξ^{2}(t−s)}f̂(s, ξ) ds |^{2}

by the Plancherel theorem. We see that if in the integration in s we bound the f̂(s, ξ) term by the finite number ∥f∥_{C(I,L^{2})}, then the ξ^{2} factor exactly cancels, and we end up with a finite value for the W_{2}^{2} norm of T f. This is, of course, wishful thinking. The assumption that f ∈ C(I, L^{2}) means that if one first takes the L^{2} norm of f̂(t, ·) for each fixed t and then takes the sup over all t, then the result is finite, while for the preceding argument to go through it is required that the L^{2} norm of the sup over t of |f(t, ·)| be finite.

We now show that any intended proof of MR for the Laplacian on (W_{p}^{2}, L^{p}), 1 ≤ p ≤ 2, must fail. Suppose otherwise. Then with T defined as above, the map ∂_{x}^{2} ∘ T : C(I, L^{p}) → C(I, L^{p}) is continuous, so by the Hausdorff–Young theorem we obtain a continuous map S : C(I, L^{p}) → C(I, L^{p′}) by composing with the Fourier transform:

Sf(t, ξ) = ∫_{0}^{t} ξ^{2}e^{−ξ^{2}(t−s)}f̂(s, ξ) ds

where p′ is the Hölder dual exponent of p, i.e.

\frac{1}{p} + \frac{1}{p′} = 1

By duality we obtain the continuous map S′ : Meas(I, L^{p}) → Meas(I, L^{p′}) given by

S′dμ(t, ξ) = ∫_{t}^{T} ξ^{2}e^{ξ^{2}(t−s)}d̂μ(s, ξ)

Here, we have assumed I = [0, T]. Restrict S′ to the closed subspace L^{1}(I, L^{p}) of the space of L^{p}–valued measures on I. For any f ∈ L^{1}(I, L^{p}), the number

∫_{0}^{T} dt ∥∫_{t}^{T} ξ^{2}e^{ξ^{2}(t−s)}f̂(s, ξ) ds∥_{L^{p′}(dξ)}

must be finite. Choose f(t, ξ) = δ(t − T)ϕ(ξ), with ϕ ∈ L^{p}. The preceding number becomes

∫_{0}^{T} dt ∥ξ^{2}e^{ξ^{2}(t−T)}ϕ̂(ξ)∥_{L^{p′}(dξ)}

Change the variable in the t integral to T − t to get

∫_{0}^{T} dt (∫_{ℝ} dξ |ξ^{2}e^{−ξ^{2}t}ϕ̂(ξ)|^{p′})^{1/p′}

The change of variable ω = ξ\sqrt{t} yields

∫_{0}^{T} dt t^{−(1+ \frac{1}{2p′})} (∫_{ℝ} dω |ω^{2}e^{−ω^{2}} ϕ̂ (\frac{ω}{\sqrt{t}})|^{p′} )^{1/p′}

Restrict the domain of integration in the inner integral to, say, the interval ω ∈ [1, 2]. On this interval, the function ω^{2}e^{−ω^{2}} is bounded away from zero, so we obtain a lower bound by a nonzero constant times

∫_{0}^{T} dt t^{−(1+ \frac{1}{2p′})} (∫_{1}^{2} dω |ϕ̂ (\frac{ω}{\sqrt{t}})|^{p′} )^{1/p′}

Changing the variable back from ω to the original ξ, we get

∫_{0}^{T} dt t^{−1} (∫_{1/\sqrt{t}}^{2/\sqrt{t}} dξ |ϕ̂(ξ)|^{p′})^{1/p′}

Split the integral into a sum, assuming for the purposes of obtaining a lower bound that T = 2^{−2N} for some integer N:

∑_{n>N} ∫_{2^{−2n}}^{2^{−2(n−1)}} dt t^{−1} (∫_{1/\sqrt{t}}^{2/\sqrt{t}} dξ |ϕ̂(ξ)|^{p′})^{1/p′}

Assume now that |ϕ̂(ξ)| is a decreasing function of |ξ|. The preceding sum is greater than

∑_{n>N} ∫_{2^{−2n}}^{2^{−2(n−1)}} dt 2^{2n} (∫_{2^{n}}^{2^{n+1}} dξ |ϕ̂(ξ)|^{p′})^{1/p′} = \frac{1}{4} ∑_{n>N} (∫_{2^{n}}^{2^{n+1}} dξ |ϕ̂(ξ)|^{p′})^{1/p′}

The latter sum is in turn no less than

∑_{n>N} 2^{n/p′}|ϕ̂(2^{n+1})|

which, up to a multiplicative constant, equals

∑_{n>N+1} 2^{n/p′}|ϕ̂(2^{n})|

Suppose we let

ϕ̂(ξ) = (1 + |ξ|)^{−n/p′} log^{−α}(1 + |ξ|)

for some \frac{1}{p′} < α ≤ 1. Then the sum above diverges, while the L^{p′} norm of ϕ̂ is finite. By a theorem of Hardy and Littlewood (theorem 82 in [34]), our choice of ϕ̂ is actually the Fourier transform of a function ϕ in L^{p}, provided we constrain the value of α somewhat further: \frac{1}{p} < α ≤ 1. This theorem does not apply in the cases p = 1, p = 2 but those cases are easy to treat directly. Thus, contrary to our assumption, S′ does not map L^{1}(I, L^{p}) into L^{1}(I, L^{p′}) and hence MR must fail for the Laplace operator on (W_{p}^{2}, L^{p}) as claimed.

2.2 Definition and basic properties of the

spaces E^{h}

In order to circumvent the obstacles to maximal regularity associated with the usual Sobolev spaces, as described in the preceding section, we construct in this section more suitable phase spaces in terms of decay properties of Fourier transforms. Related spaces have been used previously by Bricmont, Kupiainen, and Lin in their study [7] of asymptotic behavior of nonlinear parabolic equations. Definition 2.2.1. For h ∈ ℝ, let E^{h} denote the space of distributions f : S^{1} → ℝ (of course, we really mean f ∈ C^{∞}(S^{1}, ℝ)′ ) whose Fourier transform f̂: ℤ → ℝ satisfies f̂_{k} = o(|k|^{−h}) as |k| → ∞. E^{h} is a Banach space with the norm

||f||_{E^{h}} = sup_{k∈ℤ}(1 + |k|^{h})|f̂_{k}|

We will use the notation E^{∞} to refer to the inductive limit of the spaces E^{h} as h → ∞. This is simply the standard space of test functions C^{∞} with the Fréchet structure induced by the norms ∥ ∥_{E^{h}} for h ∈ ℤ^{+}; in other words, a sequence converges in E^{∞} if and only if it converges in all the spaces E^{h} with h ∈ ℤ^{+}.

A norm equivalent for convergence purposes to that defined in Eq. 2.2.1 is

∥f∥_{E^{h}}^{(2)} = max (|f̂_{0}|, sup_{k∈ℤ}|k|^{h}|f̂_{k}|)

Indeed, we have

∥ · ∥_{E^{h}}^{(2)} ≤ ∥ · ∥_{E^{h}} ≤ 2 ∥ · ∥_{E^{h}}^{(2)}

Notice that the ∥ ∥_{E^{h}}^{(2)}–norm of f is the least real number C such that |f̂_{0}| ≤ C and |f̂_{k}| ≤ C|k|^{−h} for each k ∈ ℤ, k ≠ 0.

Observe also that the definition of E^{h} amounts to the statement that (E^{h}, ∥ · ∥_{E^{h}}) is linearly isometric to the Banach space c_{0} of scalar sequences vanishing at infinity with the supremum norm, via the map f ↦ (|k|^{h}f̂_{k})_{k∈ℤ}. Among other things, this implies that the set of trigonometric polynomials is dense in E^{h}, since the set of sequences which vanish on all except a finite number of coordinates is dense in c_{0}. Also, the Banach dual of E^{h} is linearly isometric to the space ℓ^{1} of absolutely summable sequences, via the map g ↦ (|k|^{−h}ĝ_{k})_{k∈ℤ}.

Inclusion properties

We will frequently use the following inclusion properties of the spaces E^{h}.

Proposition 2.2.1. (1) E^{h} is continuously contained in the Sobolev space W^{h− \frac{1}{2} −ϵ,2} for all ϵ > 0. More generally, if 1 ≤ p ≤ 2 then there is a continuous inclusion E^{h} ⊆ W^{h− \frac{1}{p} −ϵ,p′}, where \frac{1}{p} + \frac{1}{p′} = 1.

(2) E^{h} is compactly contained in C^{h−1−ϵ} for all ϵ > 0.

(3) C^{h} is continuously contained in E^{h} if h ∈ ℤ, and C^{h} is compactly contained in E^{h−ϵ} for all ϵ > 0 for any h ∈ ℝ^{+}.

(4) E^{h} is compactly contained in E^{h−ϵ} for all ϵ > 0.

Proof.

(1) follows from the fact that all o(|k|^{− \frac{1}{2} −ϵ}) sequences are square summable. The generalization to 1 ≤ p ≤ 2 is a consequence of the Hausdorff–Young theorem, stating that the Fourier transform is a continuous mapping from L^{p} to L^{p′} for 1 ≤ p ≤ 2.

(2) follows from (1) and from the Sobolev–Rellich–Kondrachov embedding theorem, which implies that W^{μ+ \frac{1}{2} +ϵ,2} is compactly contained in C^{μ} for any μ > 0.

(3) is implied by the observation in chapter 1, section 4.6 of [18], that

f̂_{k} = C ∫_{−π}^{π} f(x)e^{−ik(x+π/k)} dx = C′ ∫_{−π}^{π} (f(x + \frac{π}{k}) − f(x)) e^{−ikx} dx,

applied with f equal to the ⌊h⌋–th derivative of an arbitrary C^{h} function g. The case h ∈ ℤ^{+} follows by (differentiation or antidifferentiation, and) the Riemann– Lebesgue lemma, i.e. the fact that L^{1} functions on the circle have Fourier transforms which belong to c_{0}.

(4) is equivalent to the statement that the set of sequences (x_{k})_{k∈ℤ} which decay faster than some predetermined negative power of |k| is compactly contained in the sequence space c_{0}. Assume (f^{(n)})_{n∈ℕ} is a bounded sequence in E^{ϵ}, and let x_{k}^{(n)} denote the k–th Fourier coefficient of f^{(n)}. Choose M ∈ ℝ^{+} such that |x_{0}^{(n)}| ≤ M and |x_{k}^{(n)}| ≤ M|k|^{−ϵ} for all k ≠ 0 in ℤ and all n ∈ ℕ. Given any positive integer K, the sequence in ℝ^{2K+1} whose n–th term has coordinates |k|^{ϵ}x_{k}^{(n)}, |k| ≤ K, is bounded in ℝ^{2K+1} and so has a subsequence which converges in ℝ^{2K+1}. By diagonalization we therefore obtain a subsequence of (x^{(n)})_{n∈ℕ} which converges coordinatewise (each k) to some sequence x = (x_{k})_{k∈ℤ}. One sees easily that |x_{0}| ≤ M and |x_{k}| ≤ M|k|^{−ϵ} for all k ≠ 0, so that x is the Fourier transform of a function f which belongs to E^{ϵ}, and similarly that along this subsequence x^{(n)} → x in c_{0}, that is, f^{(n)} → f in E^{0}, as desired. □

Comments. Above, notice that (1) implies in particular that if h > 1/2, then E^{h} is a space of true measurable functions on S^{1}. Also, the proof of (3) actually shows that the Sobolev space W^{h,1} of distributions with L^{1} derivatives of order up to h is contained in E^{h} if h ∈ ℤ.

Interpolation

Another simple property of the E^{h} spaces involves interpolation between spaces of curves in different E^{h} spaces. We will use this property in proving maximal regularity for the Mullins–Sekerka vector field in the following chapter. We include below an interpolation estimate for E^{h} norms also; the statement that E^{a} ∩ E^{b} ⊆ E^{c} whenever c is between a and b is included for clarity, but it is obvious that the left hand side of this inclusion is equal to E^{max(a,b)}.

Proposition 2.2.2. For any 0 ≤ μ ≤ 1,

1. E^{a} ∩ E^{b} is continuously contained in E^{μa+(1−μ)b}, and

||f||_{E^{μa+(1−μ)b}} ≤ ||f||_{E^{a}}^{μ}||f||_{E^{b}}^{1−μ}  (2.1)

2. C^{α}(I, E^{a})∩C^{β}(I, E^{b}) is continuously contained in C^{μα+(1−μ)β}(I, E^{μa+(1−μ)b}) and

||f||_{C^{μα+(1−μ)β}(I, E^{μa+(1−μ)b})} ≤ ||f||_{C^{α}(I,E^{a})}^{μ}||f||_{C^{β}(I,E^{b})}^{1−μ}  (2.2)

Proof. Suppose given f ∈ E^{a} ∩ E^{b}. By definition,

||f||_{E^{μa+(1−μ)b}} = max {sup{|k|^{μa+(1−μ)b}|f̂_{k}|, k ∈ ℤ}, |f̂_{0}|}  (2.3)

For each k ∈ ℤ, write

|k|^{μa+(1−μ)b}|f̂_{k}| = (|k|^{a}|f̂_{k}|)^{μ} (|k|^{b}|f̂_{k}|)^{1−μ} ≤ ||f||_{E^{a}}^{μ}||f||_{E^{b}}^{1−μ}  (2.4)

Clearly also

|f_{0}| = |f_{0}|^{μ}|f_{0}|^{1−μ} ≤ ||f||_{E^{a}}^{μ}||f||_{E^{b}}^{1−μ}  (2.5)

The first part of the statement of the proposition now follows. The second part is no more difficult to prove. Assume f ∈ C^{α}(I, E^{a})∩C^{β}(I, E^{b}). For any t, s ∈ I, we have by the first part of the proposition,

||f(t) − f(s)||_{E^{μa+(1−μ)b}} ≤ ||f(t) − f(s)||_{E^{a}}^{μ}||f(t) − f(s)||_{E^{b}}^{1−μ}  (2.6)

By assumption, the first term on the right satisfies

||f(t) − f(s)||_{E^{a}}^{μ} ≤ C_{α}^{μ}|t − s|^{αμ}  (2.7)

where C_{α} is the C^{α}(I, E^{a}) norm of f. Similarly, for the second term we have

||f(t) − f(s)||_{E^{b}}^{1−μ} ≤ C_{β}^{1−μ}|t − s|^{β(1−μ)},  (2.8)

where C_{β} is the C^{β}(I, E^{b}) norm of f, so that

||f(t) − f(s)||_{E^{μa+(1−μ)b}} ≤ C_{α}^{μ}C_{β}^{1−μ}|t − s|^{μα+(1−μ)β}  (2.9)

Finally, since by the first part we also have the “C^{0} estimate”

||f(t)||_{E^{μa+(1−μ)b}} ≤ ||f(t)||_{E^{a}}^{μ}||f(t)||_{E^{b}}^{1−μ} ≤ C_{α}^{μ}C_{β}^{1−μ} ∀t ∈ I,  (2.10)

we conclude that

||f||_{C^{μα+(1−μ)β}(I, E^{μa+(1−μ)b})} ≤ C_{α}^{μ}C_{β}^{1−μ}  (2.11)

This completes the proof of the proposition. □

2.3 Analyticity in E^{h} of multiplication and of

composition with analytic functions

The first nontrivial result involving the E^{h} spaces we shall need is the following one concerning products of functions belonging to different spaces E^{h}. It will allow us to show that composition with an analytic operation does not affect E^{h} membership. In this section we also show that the E^{h} spaces are not invariant under C^{∞} changes of variables.

Products of E^{h} functions

Theorem 2.3.1. Multiplication, (f, g) ↦ fg, is an analytic mapping E^{a}×E^{b} → E^{min(a,b)} for all pairs (a, b) satisfying a, b ≥ 0, max(a, b) > 1.

To establish the theorem we will use the following elementary result.

Lemma 2.3.2. Suppose ϕ : ℝ → ℝ is convex and non–negative. Then for any a,b in ℝ, we have

∑_{a≤k≤b, k∈ℤ}ϕ(k) ≤ ∫_{a}^{b} ϕ(s)ds + ϕ(a) + ϕ(b)

Proof. If the convex function ϕ is monotone increasing on the interval [a, b], then each term k in the sum is dominated by the integral over the associated subinterval [k, k + 1]:

ϕ(k) ≤ ∫_{k}^{k+1} ϕ(s)ds  (2.12)

Therefore, in this case

∑_{a≤k≤b, k∈ℤ}ϕ(k) ≤ ∫_{k}^{k+1} ϕ(s)ds + ϕ(b)  (2.13)

A similar statement, with ϕ(b) replaced by ϕ(a) on the right hand side, holds if ϕ is monotone decreasing. In the general case, split the interval [a, b] into two pieces [a, c] and (c, b] determined by the smallest point c at which ϕ attains its minimum. The “left sum” satisfies

∑_{a≤k≤c, k∈ℤ}ϕ(k) ≤ ∫_{a}^{⌊c⌋} ϕ(s)ds + ϕ(a)  (2.14)

The “right sum” satisfies

∑_{c<k≤b, k∈ℤ}ϕ(k) ≤ ∫_{⌈c⌉}^{b} ϕ(s)ds + ϕ(b)  (2.15)

This implies the statement of the lemma, since by the non–negativity assumption on ϕ, the integral of ϕ over the “missing interval” [⌊c⌋, ⌈c⌉] is non–negative. □

Proof of the theorem. Since multiplication is bilinear, it is enough to prove continuity of the multiplication map.

Let f ∈ E^{a}, g ∈ E^{b}, with a, b ≥ 0 and max(a, b) > 1. Then

(f̂g)_{k} = ∑_{l∈ℤ} f̂_{l}ĝ_{k−l}

so that

|(f̂g)_{k}| ≤ ∥f∥_{E^{a}}∥g∥_{E^{b}} ( ∑_{l∈ℤ,l≠0,k}|l|^{−a}|k − l|^{−b} + |k|^{−b} + |k|^{−a}) (2.16)

There is no loss of generality in assuming k ≥ 0, since if k < 0 we have:

\begin{aligned}∑_{l∈ℤ,l≠0,k}|l|^{−a}|k − l|^{−b} & = ∑_{l∈ℤ,l≠0,k}|l|^{−a}| − k + l|^{−b} \\ & = ∑_{l∈ℤ,l≠0,k}| − l|^{−a}| − k − l|^{−b} \\ & = ∑_{l∈ℤ,l≠0,k}|l|^{−a}|k′ − l|^{−b} \end{aligned},

which is the sum in Eq. 2.16 for the positive value k′.

Thus, we assume hereafter that k ≥ 0. For k = 0 we use the assumption that a + b > 1 to obtain the following bound:

\begin{aligned}|(f̂g)_{0}| & ≤ ∑_{l∈ℤ}|f̂_{l}ĝ_{−l}| ≤ ∥f∥_{E^{a}}∥g∥_{E^{b}} ∑_{l∈ℤ}(1 + |l|^{a})^{−1}(1 + |l|^{b})^{−1} \\ & ≤ C∥f∥_{E^{a}}∥g∥_{E^{b}} \end{aligned}

Now assume that k > 0. We split the infinite sum on the right-hand side of Eq. 2.16 into three parts:

∑_{l∈ℤ,l≠0,k}|l|^{−a}|k − l|^{−b} = ∑_{l<0} |l|^{−a}|k − l|^{−b} + ∑_{0<l<k} |l|^{−a}|k − l|^{−b} + ∑_{l>k} |l|^{−a}|k − l|^{−b} (2.17)

First we bound the first and third sums on the right, using Lemma 2.3.2 in the limit as one of the endpoints of the interval of integration tends to infinity (we use the assumption that a, b ≥ 0 here). For the first sum we have:

∑_{l<0} |l|^{−a}|k − l|^{−b} = ∑_{l>0} l^{−a}(k + l)^{−b} ≤ (k + 1)^{−b} + ∫_{1}^{∞} s^{−a}(k + s)^{−b}ds(2.18)

The integral on the right-hand side equals

∫_{1}^{∞} s^{−a}(k + s)^{−b}ds = k^{−a−b+1} ∫_{1/k}^{∞} s^{−a}(1 + s)^{−b}ds  (2.19)

The latter integral is convergent at ∞ if a + b > 1, and as 1/k → 0 it behaves like k^{−a+1} if a ≠ 1 and like log k if a = 1. More precisely, if a + b > 1 then we have, for a ≠ 1:

\begin{aligned}∫_{1/k}^{∞} s^{−a}(1 + s)^{−b}ds & = ∫_{1/k}^{1} s^{−a}(1 + s)^{−b}ds + ∫_{1}^{∞} s^{−a}(1 + s)^{−b}ds \\ & ≤ \frac{k^{a−1} − 1}{a − 1} + \frac{1}{a + b − 1} \\ & = \frac{k^{a−1}}{a − 1} − \frac{b}{(a − 1)(a + b − 1)} \end{aligned} (2.20)

while for a = 1, we obtain:

\begin{aligned}∫_{1/k}^{∞} s^{−a}(1 + s)^{−b}ds & = ∫_{1/k}^{1} s^{−1}(1 + s)^{−b}ds + ∫_{1}^{∞} s^{−1}(1 + s)^{−b}ds \\ & ≤ log k + \frac{1}{b} \end{aligned} (2.21)

Therefore, if a ≠ 1, the integral in Eq. 2.19 is bounded above by:

\frac{k^{−b}}{a − 1} − \frac{bk^{−a−b+1}}{(a − 1)(a + b − 1)}  (2.22)

while if a = 1, the integral in Eq. 2.19 is bounded by:

k^{−b} log k + \frac{1}{b}k^{−b}  (2.23)

Notice that the assumption max(a, b) > 1 implies that the exponent −a − b + 1 in Eq. 2.22 is less than min(−a, −b). Also, if a = 1 then b > 1, and so in Eq. 2.23 the exponent −b is less than −a. Thus, from Eq. 2.18 we conclude that the first sum on the right-hand side in Eq. 2.17 is bounded above by a constant times k^{− min(a,b)}. The third sum in Eq. 2.17 is reduced by the change of variables l = k − l′ to the sum in Eq. 2.18, and therefore is also bounded by a constant multiple of k^{− min(a,b)}.

Next we bound the middle sum in Eq. 2.17, again using the preceding Lemma:

\begin{aligned}∑_{0<l<k} l^{−a}(k − l)^{−b} & ≤ ∫_{1}^{k−1} s^{−a}(k − s)^{−b}ds + (k − 1)^{−b} + (k − 1)^{−a} \\ & = k^{−a−b+1} ∫_{1/k}^{1−1/k} s^{−a}(1 − s)^{−b}ds + (k − 1)^{−b} + (k − 1)^{−a} \end{aligned} (2.24)

Split the interval of integration into two subintervals meeting at 1/2. We get, assuming for the moment that 0 < a, b ≠ 1:

\begin{aligned}∫_{1/k}^{1−1/k} s^{−a}(1 − s)^{−b}ds & ≤ 2^{b} ∫_{1/k}^{1/2} s^{−a}ds + 2^{a} ∫_{1/2}^{1−1/k}(1 − s)^{−b}ds \\ & = \frac{2^{b}}{a − 1}(k^{a−1} − 2^{a−1}) + \frac{2^{a}}{b − 1}(k^{b−1} − 2^{b−1}) \end{aligned}(2.25)

If a = 1, then instead we have:

\begin{aligned}∫_{1/k}^{1−1/k} s^{−a}(1 − s)^{−b}ds & ≤ 2^{b} ∫_{1/k}^{1/2} s^{−1}ds + 2 ∫_{1/2}^{1−1/k}(1 − s)^{−b}ds \\ & = 2^{b}(log k − log 2) + \frac{2}{b − 1}(k^{b−1} − 2^{b−1}) \end{aligned} (2.26)

The case b = 1 is essentially identical. Finally, consider the case a = 0 (again, b = 0 is similar). Then we have:

\begin{aligned}∫_{1/k}^{1−1/k} s^{−a}(1 − s)^{−b}ds & = ∫_{1/k}^{1−1/k}(1 − s)^{−b}ds \\ & ≤ \frac{1}{b − 1}(k^{b−1} − 1) \end{aligned}  (2.27)

From Eqs. 2.24, 2.25, 2.26, and 2.26, and using the assumption that max(a, b) > 1, we infer that the middle sum in Eq. 2.17 is bounded above by a constant times k^{− min(a,b)}, and therefore we have:

|(f̂g)_{k}| ≤ C∥f∥_{E^{a}}∥g∥_{E^{b}}|k|^{−a∧b}

This completes the required estimate for nonzero values of k. By the equivalence of the norms ∥ ∥_{E^{h}} and ∥ ∥_{E^{h}}^{(2)}, this completes the proof. □

Analytic images of E^{h} functions

Theorem 2.3.3. Let Φ : S → ℝ be analytic on some open set S in ℝ. Then composition with Φ, f ↦ Φ ∘ f, is an analytic mapping from the open subset of E^{h} consisting of functions f whose image is contained in S, to E^{h} itself, for any h > 2.

The proof of this Theorem requires the following analogous result for the space C^{μ} of Hölder continuous functions with exponent μ.

Lemma 2.3.4. If Φ is as in the statement of Theorem 2.3.3, then the mapping f ↦ Φ ∘ f is analytic from the open set in C^{μ} consisting of functions whose range is contained in S, to C^{μ}, for any μ ≥ 0.

Proof of the Lemma. For each y_{0} ∈ S we have:

Φ(y) = ∑_{n∈ℕ} \frac{D^{n}Φ}{n!}(y_{0})(y − y_{0})^{n}  (2.28)

The series in Eq. 2.28 converges for all y ∈ ℂ with |y| < r(y_{0}). Now fix f ∈ C^{μ} with f(S^{1}) ⊂ S. Then for g ∈ C^{μ} we have formally for all x ∈ S^{1}:

Φ(f(x) + g(x)) = ∑_{n∈ℕ} \frac{D^{n}Φ}{n!}(f(x))g(x)^{n}  (2.29)

For fixed x, this series converges in ℝ if |g(x)| < r(f(x)). However, since the range of f is compact and connected, for some R_{0} the range of f is contained in the union of a finite number of disks of radius 2R_{0} contained in S and we have:

inf_{x∈S^{1}} r(f(x)) > 2R_{0} > 0.  (2.30)

Therefore, the series in Eq. 2.29 converges for all x ∈ S^{1} assuming ∥g∥_{C^{0}} < R_{0}. We will show that for such g, the series in fact converges in the space C^{μ}, thus proving the Lemma.

First observe that for each m ∈ ℕ, one has an analogous series for D^{m}Φ, which applied to y = f(x) + g(x) yields:

D^{m} (Φ(f(x) + g(x))) = ∑_{n∈ℕ} \frac{D^{n+m}Φ}{n!}(f(x))g(x)^{n}  (2.31)

Using Cauchy’s integral theorem for analytic functions [1], it follows that the radius of convergence r^{(m)}(f(x)) for the m–th derivative series in Eq. 2.31 equals the radius of convergence r(f(x)) for the series in Eq. 2.29, and that one has the following spatially uniform Cauchy estimate for the derivatives:

|\frac{D^{n}Φ}{n!}(y)| ≤ ∥Φ∥_{C^{0}(B_{R_{0}}(f(S^{1})))}R_{0}^{−n} ∀y ∈ B_{R_{0}}(f(S^{1})), n ∈ ℕ (2.32)

We will use the notation |Φ|_{C^{n}} for the supremum of the absolute value of (D^{n}Φ)(y) over all points y in the R_{0}–neighborhood B_{R_{0}}(f(S^{1})) of the range of f. When n = 0, the seminorm |Φ|_{C^{0}} is equal to the usual norm ∥Φ∥_{C^{0}}.

Cauchy’s estimate implies that for any R < R_{0}:

∑_{n∈ℕ} \frac{1}{n!}|Φ|_{C^{n+m}}R^{n} < ∞ ∀m ∈ ℕ  (2.33)

In particular, the problem of proving the convergence of the original series 2.29 in the space C^{μ} is reduced to a similar problem for the series 2.31 in the space C^{μ−m} if m ∈ ℕ with 0 ≤ μ. It follows that no generality is lost by assuming μ ∈ [0, 1).

We now show for μ ∈ [0, 1) that the series in Eq. 2.29 converges in the space C^{μ} if ∥g∥_{C^{μ}} < R_{0}.

Bounding the C^{0} norm of the terms of the series in Eq. 2.29 is simple:

∥\frac{D^{n}Φ}{n!}(f)g^{n}∥_{C^{0}} ≤ \frac{1}{n!}∥Φ∥_{C^{n}}∥g∥_{C^{0}}^{n}  (2.34)

By Eq. 2.33 with m = 0, the right-hand side of this inequality is the n–th term of a convergent series if ∥g∥_{C^{0}} < R_{0}.

Now consider the C^{μ} seminorm, μ ∈ (0, 1), of the n–th term in Eq. 2.29. Choose x and x + y in S^{1} (meaning [−π, π)). We may write the difference

\frac{D^{n}Φ}{n!}(f(x + y))g(x + y)^{n} − \frac{D^{n}Φ}{n!}(f(x))g(x)^{n}  (2.35)

as the sum of two terms:

\frac{D^{n}Φ}{n!}(f(x + y)) (g(x + y)^{n} − g(x)^{n}) + (\frac{D^{n}Φ}{n!}(f(x + y)) − \frac{D^{n}Φ}{n!}(f(x))) g(x)^{n} (2.36)

For the first term:

\begin{aligned}|\frac{D^{n}Φ}{n!}(f(x + y)) (g(x + y)^{n} − g(x)^{n})| & ≤ \frac{1}{n!}|Φ|_{C^{n}} |∫_{g(x)}^{g(x+y)} nz^{n−1}dz| \\ & ≤ \frac{n}{n!}|Φ|_{C^{n}} |g(x + y) − g(x)| ∥g∥_{C^{0}}^{n−1} \\ & ≤ \frac{n}{n!}|Φ|_{C^{n}}∥g∥_{C^{0}}^{n−1}∥g∥_{C^{μ}}|y|^{μ} \end{aligned} (2.37)

For the second term:

\begin{aligned}|(\frac{D^{n}Φ}{n!}(f(x + y)) − \frac{D^{n}Φ}{n!}(f(x))) g(x)^{n}| & ≤ \frac{1}{n!}|Φ|_{C^{n+1}} |f(x + y) − f(x)| ∥g∥_{C^{0}}^{n} \\ & ≤ \frac{1}{n!}|Φ|_{C^{n+1}}∥g∥_{C^{0}}^{n}∥f∥_{C^{μ}}|y|^{μ} \end{aligned} (2.38)

From Eqs. 2.34, 2.37, and 2.38, we obtain:

∥\frac{D^{n}Φ}{n!}(f)g^{n}∥_{C^{μ}} ≤  (2.39)

\frac{1}{n!}|Φ|_{C^{n}}∥g∥_{C^{0}}^{n} + \frac{n}{n!}|Φ|_{C^{n}}∥g∥_{C^{μ}}^{n} + \frac{1}{n!}|Φ|_{C^{n+1}}∥g∥_{C^{0}}^{n}∥f∥_{C^{μ}}

By Eq. 2.33 with m = 0 and m = 1, we see that the right-hand side of Eq. 2.39 is the n–th term of a convergent series if ∥g∥_{C^{μ}} < R_{0}. This proves the Lemma.

□

Proof of Theorem 2.3.3. Differentiate the composition Φ ∘ f twice to get:

(Φ ∘ f)′′ (x) = Φ′′(f(x))f′(x)^{2} + Φ′(f(x))f′′(x)  (2.40)

By the embedding results proved above in Proposition 2.2.1, we have f ∈ C^{h−1−ϵ}. Therefore, by the preceding composition lemma for Hölder spaces, Φ′′ ∘ f ∈ C^{h−1−ϵ}, with analytic dependence on f. Embedding in reverse, Φ′′ ∘f ∈ E^{h−1−2ϵ}. By the Product Theorem 2.3.1, the product Φ′′(f(x))f′(x)^{2} belongs to E^{h−1−2ϵ}, with analytic dependence on f; also, the term Φ′(f(x))f′′(x) belongs to E^{h−2}. In this way we conclude that the right-hand side belongs to E^{h−2} if h > 2, with analytic dependence on f. This completes the proof of the Theorem. □

Comments. Observe from the preceding proof that a similar result holds whenever Φ ∈ C^{μ} with μ > h > 2. One finds for such Φ that for any f ∈ E^{h}, the composition Φ ∘ f belongs to E^{h}. The smoothness of the composition as a function of f is then only of finite order, of course.

Parametrization dependence of the E^{h} spaces

In contrast to the above, precomposition (“composition on the right”) with a C^{∞} function is not well behaved in the spaces E^{h}. The argument given below involves van der Korput’s sharp asymptotic lemma. As a consequence of this, the E^{h} spaces are not invariant under C^{∞} changes of variables. Therefore, these spaces cannot be defined invariantly on C^{∞} manifolds. More concretely, one is required to specify the parametrizations used for the various curves when stating assumptions or conclusions regarding E^{h} membership.

Let ϕ ∈ C^{∞}(S^{1}). We identify ϕ in the natural manner with a function defined on [−π, π]. We seek to understand the behavior as k → ∞ of the Fourier coefficients f ∘ ϕ^{−1}, where f ∈ E^{h}. Using the Parseval identity we have:

(f ∘ ϕ^{−1})_{k}^{∧} = ∫_{−π}^{π} f(y) e^{−ikϕ(y)} ϕ′(y) dy  (2.41)
= ∑_{l∈ℤ}(ϕ′f)_{l}^{∧} ∫_{−π}^{π} e^{i(ly−kϕ(y))} dy  (2.42)
= ∑_{l∈ℤ}(ϕ′f)_{l}^{∧} ∫_{−π}^{π} e^{ik( \frac{l}{k} y−ϕ(y))} dy  (2.43)

The terms (ϕ′f)_{l}^{∧} decay like |l|^{−h}. The integral is an oscillatory integral for which the asymptotic behavior for large k can be found by a sharp stationary phase estimate due to van der Korput (see [31]). Since ϕ is continuous on the circle S^{1}, it must have at least two critical points. For the purpose of exhibiting a counterexample to the conjecture that the composition f ∘ ϕ^{−1} belongs to E^{h}, assume that the function ϕ(y) has exactly two critical points p and q, at both of which the second differential of ϕ is nondegenerate, and such that

ϕ′′(p) + ϕ′′(q) ≠ 0  (2.44)

One may consider the concrete example ϕ(y) = e^{sin y}. Then van der Korput’s lemma states that for any ψ ∈ C_{0}^{∞}(−π, π) with sufficiently small support near the points p and q, and with ψ(p) ≠ 0, ψ(q) ≠ 0, the integral

∫_{−π}^{π} e^{ik(−ϕ(y))}ψ(y) dy  (2.45)

decays like A(ϕ′′(p) + ϕ′′(q))|k|^{−1/2}, where A is a constant. We will denote the product A(ϕ′′(p) + ϕ′′(q)) as B(0). The lemma also states that if, on the other hand, p and q lie outside the support of ψ, then the integral in eq. 2.45 decays faster than any negative power of k.

Assume ψ is supported near p and q. If c is a number with

\frac{1}{e} = min ϕ′ ≤ c ≤ max ϕ′ = e,  (2.46)

then the new function ϕ(y) − cy will, like ϕ(y) itself, have two critical points p(c) and q(c), which will determine the asymptotic behavior of the oscillatory integral

∫_{−π}^{π} e^{ik(cy−ϕ(y))}ψ(y) dy  (2.47)

More precisely, this integral will be asymptotic to B(c)|k|^{−1/2}, where B(c) denotes the product A(ϕ′′(p(c)) + ϕ′′(q(c))). All numbers c outside the range in eq. 2.46 will lead to decay of the corresponding integral at a rate faster than |k|^{−n} for any n ∈ ℕ.

Now apply these decay estimates to the integrals which appear in eq. 2.43. Split the set of integrals into two parts, according to the value of \frac{l}{k}:

(f ∘ ϕ^{−1})_{k}^{∧} = ∑_{\frac{1}{e} ≤ \frac{l}{k} ≤e} (ϕ′f)_{l}^{∧} ∫_{−π}^{π} e^{ik( \frac{l}{k} y−ϕ(y))} dy  (2.48)
+ ∑_{other values of l} (ϕ′f)_{l}^{∧} ∫_{−π}^{π} e^{ik( \frac{l}{k} y−ϕ(y))} dy  (2.49)

The integrals belonging to the second sum on the right decay superalgebraically. It is easy to see that their decay rate is superalgebraic in (min |ϕ′ − \frac{l}{k}|)^{−1} as well (see the proof of Proposition 1 in Stein’s article in [31]). It follows that the second sum on the right above converges and decays superalgebraically in k. We should therefore focus our attention on the first sum, which by van der Korput’s lemma is asymptotic to

∑_{\frac{1}{e} ≤ \frac{l}{k} ≤e} (ϕ′f)_{l}^{∧} B (\frac{l}{k} ) |k|^{−1/2}  (2.50)

The sum of second derivatives which appears in eq. 2.44, depends continuously on ϕ in the C^{2} norm. Thus, there exists ϵ such that for all numbers c with |c| < ϵ, the function ϕ(y) − cy satisfies the asymptotic bound

| ∫_{−π}^{π} e^{ik(cy−ϕ(y))}ψ(y) dy| ≥ B′|k|^{−1/2},  (2.51)

with B′ > B/2 > 0. Since the number of indices l satisfying |\frac{l}{k} −c| < ϵ is greater than |k|ϵ, it follows that for the first sum in eq. 2.49 we have the estimate

| ∑_{a< \frac{l}{k} <b} (ϕ′f)_{l}^{∧} ∫_{−π}^{π} e^{ik( \frac{l}{k} y−ϕ(y))} dy| ≥ ∑_{l} | (ϕ′f)_{l}^{∧} |B′|k|ϵ|k|^{−1/2} (2.52)

If we now select f ∈ E^{h} so that f̂ϕ′_{k} ≡ |k|^{−(h+ϵ)} as |k| → ∞, then we will have

| (f ∘ ϕ^{−1})_{k}^{∧} | ≥ |k|^{−(h+ϵ)}B′|k|ϵ|k|^{−1/2}  (2.53)
= B′′|k|^{−(h−1/2+ϵ)}  (2.54)

Thus, f ∘ ϕ^{−1} ∉ E^{h}, as claimed.

2.4 The O(θ^{2}) lemma and E^{h} analyticity of di-

vision by vanishing denominators

Next we study division as a map on certain submanifolds of E^{h}. Theorem 2.3.3 implies that division is analytic provided the denominator is bounded away from 0. In examining potential theory in the next chapter, we will need to consider quotients of the special type described in the following result, with both numerator and denominator vanishing quadratically at a common point.

Theorem 2.4.1. Let a > 2m + 1 for some positive integer m. Define

O_{a}(x^{2m}) = {f ∈ E^{a}|f(x) = O(x^{2m}) as x → 0},

Θ_{a}(x^{2m}) = \{f ∈ O_{a}(x^{2m})| \frac{1}{f(x)} ∈ O_{a}(x^{−2m}) as x → 0 and f(x) ≠ 0 ∀x ≠ 0\}.

Both O_{a}(x^{2m}) and Θ_{a}(x^{2m}) are analytic submanifolds of E^{a}, locally analytically diffeomorphic to the space O_{a}(x^{2m}), with codimension 2m in E^{a}. The map (f, g) ↦ \frac{f}{g} is analytic as a map O_{a}(x^{2m}) × Θ_{a}(x^{2m}) → E^{a−2m}, and is given near each point (f, g) by a power series whose region of convergence depends only on the value of the 2m–th derivative (D^{2m}g)(0) and on the E^{a} norm of g.

Before proving analyticity of division, we show that division is well defined between the spaces given in the statement of the theorem.

Lemma 2.4.2. If f ∈ O_{a}(x^{2m}), g ∈ Θ_{a}(x^{2m}), a > 2m + 1, then the quotient f(x)/g(x) is in E^{a−2m}. Furthermore, there exist a continuous nonincreasing function Φ(·) : ℝ^{+} → ℝ and a real number C such that for all such functions f, g,

∥\frac{f}{g} ∥_{E^{a−2m}} ≤ C^{m}Φ (inf_{x} \frac{|g(x)|}{4^{m} sin^{2m}(x/2)}) ∥f∥_{E^{a}}∥g∥_{E^{a}}

Proof. Consider first the case m = 1. The function 1/2 tan(x/2) is the 2π– periodization of the function 1/x on ℝ in the sense that

∑_{k∈ℤ} \frac{1}{x − 2πk} = \frac{1}{2 tan(x/2)}  (2.55)

The sum on the left must be interpreted as a principal value sum:

∑_{k∈ℤ} \frac{1}{x − 2πk} = lim_{k_{0}→∞} ∑_{|k|≤k_{0}} \frac{1}{x − 2πk}  (2.56)
= \frac{1}{x} + lim_{k_{0}→∞} ∑_{1≤k≤k_{0}} (\frac{1}{x − 2πk} + \frac{1}{x + 2πk} )  (2.57)
= \frac{1}{x} + lim_{k_{0}→∞} ∑_{1≤k≤k_{0}} \frac{2x}{x^{2} − 4π^{2}k^{2}}  (2.58)

Eq. 2.55 follows from a theorem by Mittag–Leffler (see [1]) which implies that two meromorphic functions with identical poles and identical singular parts of their Laurent expansions at these poles can differ only by an entire function. This is clearly the case for the functions in eq. 2.55. Taking into account the periodicity in x of the functions involved, one then shows by analyzing the behavior at ±i∞ that this entire function is bounded and thus must be constant; since both functions are odd, the constant must be 0, thus proving eq. 2.55.

The standard Poisson summation argument (as in Strichartz’s book [33]) now shows that the Fourier coefficients of 1/2 tan(x/2) equal the values on the integers of the Fourier transform of 1/x. But, interpreting 1/x as a Schwartz distribution acting on test functions as a Cauchy principal value integral, we know that its Fourier transform is given by

(\frac{1}{x})^{∧} (ω) = −\frac{i}{2}sgnω

and so the Fourier coefficients of 1/2 tan(x/2) are

(\frac{1}{2 tan(x/2)})_{k}^{∧} = −\frac{i}{2}sgnk,  (2.59)

for all k ∈ ℤ. The periodization of 1/x^{2} is therefore

∑_{k∈ℤ} \frac{1}{(x − 2πk)^{2}} = −\frac{d}{dx} ∑_{k∈ℤ} \frac{1}{(x − 2πk)} = −\frac{d}{dx} \frac{1}{2 tan(x/2)} = \frac{1}{4 sin^{2}(x/2)}

and its Fourier coefficients are

(\frac{1}{4 sin^{2}(x/2)})_{k}^{∧} = \frac{k}{2}sgnk

The Fourier coefficients of the product f(x)/4 sin^{2}(x/2) are obtained through convolution:

(\frac{f(x)}{4 sin^{2}(x/2)})_{k}^{∧} = ∑_{l∈ℤ} f̂_{l}(k − l)sgn(k − l) = ∑_{l<k} f̂_{l}(k − l) − ∑_{l>k} f̂_{l}(k − l)

Using the assumption that f(0) = 0 and \frac{df}{dx}(0) = 0 (f is in C^{2} by Sobolev embedding), we have

∑_{l∈ℤ} f̂_{l} = 0 = ∑_{l∈ℤ} f̂_{l}l

and so we obtain

|(\frac{f(x)}{4 sin^{2}(x/2)})_{k}^{∧}| ≤ 2 ∑_{l≥k} (|k||f̂_{l}| + |f̂_{l}l|) = O(|k|^{−(a−2)}),

since we have assumed that f̂_{k} = O(|k|^{−a}). By writing

\frac{f(x)}{g(x)} = \frac{f(x)}{4 sin^{2}(x/2)} (\frac{g(x)}{4 sin^{2}(x/2)})^{−1}

and using the fact that the inversion map x ↦ 1/x is analytic away from the origin, we conclude that the quotient f(x)/g(x) is in E^{a−2} and a function Φ exists as described in the statement. We can take C = 1 here.

The result for general m is now obtained by induction, as follows. Assume that the lemma holds for a certain m and also for the base case m = 1. We leave the value of C unspecified for now. Suppose g(x) = O(x^{2(m+1)}) as x → 0 and f(x)/g(x) is bounded near x = 0. Write

\frac{f(x)}{g(x)} = \frac{f(x)/4 sin^{2}(x/2)}{g(x)/4 sin^{2}(x/2)}

By the case m = 1, we know that f(x)/4 sin^{2}(x/2) and g(x)/4 sin^{2}(x/2) are in E^{a−2}, and clearly f(x)/4 sin^{2}(x/2) and g(x)/4 sin^{2}(x/2) are O(x^{2m}), so we conclude by the induction hypothesis that f(x)/g(x) is in E^{(a−2)−2m} as desired, and if we let

C = max (1, Φ(1)^{2}∥4 sin^{2}(x/2)∥_{E^{1}}^{2}) ,

then the bound given in the statement is confirmed. □

Proof of the theorem. The assumption that a > 2m + 1 implies by Proposition 2.2.1 that E^{a} is continuously embedded in C^{2m}. It follows that:

O_{a}(x^{2m}) = \{f ∈ E^{a}|\frac{d^{n}f}{dx^{n}} (0) = 0 for n = 0, 1, 2, ..., 2m − 1\}

Thus, the implicit function theorem shows that O_{a}(x^{2m}) is an analytic submanifold of E^{a} with codimension 2m. One sees also that Θ_{a}(x^{2m}) is an open subset of O_{a}(x^{2m}), thus an analytic submanifold of E^{a} of the same codimension.

Now fix (f, g) ∈ O_{a} × Θ_{a}. For small enough ||δg||_{E^{a}}, we have g + δg ∈ Θ_{a}, and for any δf ∈ O_{a} we have f + δf ∈ O_{a}. Write

\frac{f + δf}{g + δg} − \frac{f}{g} = \frac{\frac{δf}{g} − \frac{f}{g} \frac{δg}{g}}{1 + \frac{δg}{g}} = (\frac{δf}{g} − \frac{f}{g} \frac{δg}{g} ) ∑_{n∈ℕ}(−\frac{δg}{g} )^{n}  (2.60)

Applying the Lemma on E^{a−2m} boundedness to each quotient, and using the Theorem on boundedness of the product as a bilinear map on E^{a−2m} ( 2.3.1), we see that the series converges in the norm of E^{a−2m} for all sufficiently small ||δg||_{E^{a}}. Furthermore, the right-hand side of Eq. 2.60 is an analytic function of (δf, δg), since the expressions

\frac{δf}{g} , \frac{f}{g} \frac{δg}{g}

are bilinear in (δf, δg), the series

∑_{n∈ℕ}(−\frac{δg}{g} )^{n}

is analytic in (δf, δg), and multiplication is an analytic map E^{a−2m} × E^{a−2m} → E^{a−2m} by Theorem 2.3.1. This proves analyticity of O(θ^{2m}) division as stated. It is clear from the above comments that the region of convergence of the power series expansion in (δf, δg) of the quotient \frac{f+δf}{g+δg} depends only on the validity of the series expansion ∑_{n∈ℕ} (−\frac{δg}{g} )^{n}. In turn, the convergence of this series depends only on ∥g∥_{E^{a}} and its validity is further constrained only by the requirement that the E^{a} norm of the perturbation δg be such that g+δg ∈ Θ_{a}. The latter condition depends only on the value of the 2m–th derivative D^{2m}g(0), which exists and depends continuously on g in the E^{a} norm by continuity of the embedding of E^{a} into C^{2m} for a > 2m + 1 as proved in Proposition 2.2.1. The proof of the Theorem is now complete. □

2.5 Integral operators with E^{h} kernels

Finally, we consider some elementary results on integral operators which will be invoked below when dealing with double-layer potentials. We are interested in operators of the form

(Kf)(θ_{1}) = ∫_{−π}^{π} k(θ_{1}, θ_{2})f(θ_{2})dθ_{2}  (2.61)

where the kernel k(θ_{1}, θ_{2}) belongs to E^{s} in the θ_{1} variable and to E^{q} in the θ_{2} variable. We need q > 1 so that E^{q} is continuously contained in C^{0}. We also assume that the E^{s} norm of k(·, θ_{2}) is bounded uniformly in θ_{2}.

Continuity

Lemma 2.5.1. K maps L^{1} to E^{s} continuously. In fact,

||Kf||_{E^{s}} ≤ ( sup_{θ_{2}∈[−π,π]} ||k(·, θ_{2})||_{E^{s}}) ||f||_{L^{1}}

Proof. We have

K̂f_{k} = ∫_{−π}^{π} k̂(·, θ_{2})_{k}f(θ_{2})dθ_{2}

so that

|K̂f_{k}| ≤ ( sup_{θ_{2}∈[−π,π]} ||k(·, θ_{2})||_{E^{s}}|k|^{−s}) ∫_{−π}^{π} |f(θ_{2})|dθ_{2}

This completes the proof. □

Notice that, since by Proposition 2.2.1 E^{b} is continuously contained in L^{1} for all b > \frac{1}{2}, Lemma 2.5.1 yields continuity of K as a map from E^{b} to E^{s} for b > \frac{1}{2}.

Differentiability

We now observe, in the lemma below, that if the kernel k depends smoothly on some parameter r ∈ E^{h}, then the associated integral operator K depends smoothly on r as well, with the same degree of smoothness. For notational simplicity we consider the behavior of the various mappings near r = 0 rather than near an arbitrary r_{0}, although clearly no generality is lost in this way. We will be interested in the dependence of (k(r))(θ_{1}, θ_{2}) on θ_{1} for fixed θ_{2}. We will assume that the map r ↦ (k(r))(·, θ_{2}) has a certain degree of differentiability, and we will conclude that the map r ↦ K(r) does also.

In this section we let D denote (Fréchet) differentiation with respect to the functional argument r. This deviates from our usual convention that D should denote differentiation with respect to the polar angle. We will abbreviate the result of applying the n–multilinear map (D^{n}k)|_{r_{0}=0} to the “diagonal” n–tuple (r, r, ..., r) as (D^{n}k)(r^{n}). Given n, we have for each fixed θ_{2} a different n– multilinear mapping (D^{n}k)(θ_{2}) : (E^{h})^{n} → E^{s}, which acts on each r ∈ E^{h} to produce a function ((D^{n}k)(θ_{2}))(r^{n}) ∈ E^{s}. We will sometimes write (D^{n}k)(r^{n})(θ_{1}, θ_{2}) when we really mean {((D^{n}k)(θ_{2}))(r^{n})}(θ_{1}).

We define the notation

|(D^{n}k)(θ_{2})|_{h,s} = sup_{r∈E^{h},∥r∥_{E^{h}}≤1} ∥ ((D^{n}k)(θ_{2})) (r^{n})∥_{E^{s}}  (2.62)

We then have for all r ∈ E^{h}:

∥(D^{n}k)(r^{n})∥_{E^{s}} ≤ |(D^{n}k)(θ_{2})|_{h,s} ∥r∥_{E^{h}}^{n}  (2.63)

Lemma 2.5.2. If k = k(r) as a map k : r ↦ k(·, θ_{2}) from E^{h} to E^{s} is C^{N} in r uniformly in θ_{2}, i.e.

k(r)(θ_{1}, θ_{2}) = ∑_{n≤N} ((\frac{D^{n}k}{n!} (θ_{2})) (r^{n})) (θ_{1}) + o(r^{N})(θ_{1}, θ_{2})

with

sup_{θ_{2}∈[−π,π]} ∥o(r^{N})(·, θ_{2})∥_{E^{s}} = o(∥r∥_{E^{h}}^{N}),

then K is of class C^{N} in r, as a map K : E^{h} → C(L^{1}, E^{s}). The conclusion holds if N is a natural number, ∞, or ω. C^{ω} means analytic. If N = ω, then we assume:

(k(r))(θ_{1}, θ_{2}) = ∑_{n∈ℕ}((\frac{D^{n}k}{n!} (θ_{2})) (r^{n})) (θ_{1}),  (2.64)

and we assume the scalar series ∑_{n∈ℕ} |(\frac{D^{n}k}{n!}(θ_{2}))|_{h,s} x^{n} has a nonzero radius of convergence which is uniform in θ_{2}.

Proof. First consider the case N < ∞. Using the assumption on k, we have

(K(r)f)(θ_{1}) = ∑_{n≤N} ∫_{−π}^{π} dθ_{2}f(θ_{2})\frac{D^{n}k}{n!} (r^{n})(θ_{1}, θ_{2}) + ∫_{−π}^{π} dθ_{2}f(θ_{2})o(r^{N})(θ_{1}, θ_{2})

Applying the preceding lemma to the last term on the right, we see that it is o(r^{N}) in the operator norm of C(L^{1}, E^{s}). This proves the statement if N < ∞, and therefore also if N = ∞.

Next we treat the case N = ω. Let f ∈ L^{1}, and consider the partial sum:

(K_{M}(r)f)(θ_{1}) = ∑_{n≤M} ∫_{−π}^{π} \{((\frac{D^{n}k}{n!} (θ_{2})) (r^{n})) (θ_{1})\} f(θ_{2}) dθ_{2} (2.65)

We wish to show that as M → ∞, the remainder K(r)−K_{M}(r) converges to 0 in the operator norm of C(L^{1}, E^{s}). The assumption in the statement regarding the uniformity of the radius of convergence of the Taylor expansion may be written as follows:

sup_{θ_{2}} lim sup_{n→∞} |(\frac{D^{n}k}{n!} (θ_{2}))|_{h,s}^{1/n} =: γ < ∞  (2.66)

Recalling the definition of | |_{h,s} from 2.62, and using Eq. 2.65, Eq. 2.66, and Lemma 2.5.1, we have for all M larger than some fixed M_{0} and any ϵ > 0:

\begin{aligned}∥(K(r) − K_{M}(r))f∥_{E^{s}} & ≤ ∑_{n>M} ∫_{−π}^{π}(γ + ϵ)^{n}∥r∥_{E^{h}}^{n}∥f∥_{L^{1}} dθ_{2} \\ & ≤ 2π∥f∥_{L^{1}} ∑_{n>M}(γ + ϵ)^{n}∥r∥_{E^{h}}^{n} \end{aligned}  (2.67)

Since this holds for all ϵ > 0, we see letting ϵ → 0 that if the E^{h} norm of r is less than 1/(γ), then the sum over n > M on the right-hand side of Eq. 2.67 is the tail of a convergent series, and therefore vanishes in the limit as M → ∞. This completes the proof. □

Compactness

For the purpose of applying the Fredholm–Riesz–Schauder theory in the following chapter, we need to prove compactness of certain integral operators.

Lemma 2.5.3. Suppose there exist h_{0} ∈ ℝ and d ∈ ℝ, with h_{0} + d > 1/2, such that the map r ↦ k(r)(·, θ_{2}) is continuous from E^{h} to E^{h+d} uniformly in θ_{2} for all h ≥ h_{0}. Then for every r ∈ E^{h}, with h ≥ h_{0}, the associated integral operator K(r) defined in Eq. 2.61 is compact as a map from L^{1} to E^{h+d}.

Proof. Fix r ∈ E^{h}, for some h ≥ h_{0}. There exists a sequence (r_{n})_{n∈ℕ} in E^{∞} which converges to r in E^{h}. By the assumption that k maps E^{a} to E^{a+d} uniformly continuously in θ_{2} for all a ≥ h_{0}, we have

k(r_{n})(·, θ_{2}) ∈ E^{∞}  (2.68)

and

lim_{n→∞} sup_{θ_{2}} ∥k(r_{n})(·, θ_{2}) − k(r)(·, θ_{2})∥_{E^{h+d}} = 0.  (2.69)

Let K_{n} = K(r_{n}). Then by Lemma 2.5.1 and Eq. 2.68, K_{n} maps L^{1} to E^{∞} continuously. By Proposition 2.2.1, the embedding of E^{∞} into E^{h+d} is compact. It follows that each K_{n} is a compact mapping from L^{1} to E^{h+d}. Furthermore, by Lemma 2.5.1 and Eq. 2.69, we have:

lim_{n→0} ∥K_{n} − K∥_{C(L^{1},E^{h+d})} = 0  (2.70)

This proves that K is compact as a map from L^{1} to E^{h+d}, as claimed. □

Analyticity of the Dirichlet to

Neumann map and maximal regularity

for the Mullins–Sekerka equation

In this chapter we prove Theorem 1.2.1. We show that for h > 4, the Dirichlet to Neumann map N(d) maps E^{h−2} to E^{h−3} analytically in d ∈ E^{h}, and that maximal regularity holds for the Mullins–Sekerka vector field N(d)κ(d) with respect to the phase pair (E^{h−3}, E^{h}), if d is sufficiently small in the space E^{h}.

The strategy is to represent the solution of the mixed boundary value problem for the Laplace operator using single-layer potentials, in terms of which the Neumann jump across the interface may be expressed rather cleanly. The single-layer integral operator S and the adjoint K^{∗} of the double-layer integral operator appear throughout. They are shown to be continuous from E^{h−3} to E^{h−2}, respectively from E^{h−3} to E^{h−3}, using the O(θ^{2}) Lemma 2.4.2. Analyticity in d is obtained from Lemma 2.5.2. The Fredholm–Riesz–Schauder theory for operators of the form identity + compact on a Banach space is used to invert the single-layer operator to obtain the Dirichlet to Neumann map. Compactness of the operators involved follows from Lemma 2.5.3.

The maximal regularity argument outlined in the Introduction and implemented in detail in the present chapter involves proving that maximal regularity holds for the circular interface case d = 0. The sectoriality of the linearized vector field is an immediate consequence of the estimate used to prove maximal regularity. Since the set of operators satisfying MR is open in C(E^{h}, E^{h−3}), MR follows for small d. This yields well–posedness, as desired. The E^{h} spaces allow a fairly simple proof of maximal regularity; most of the work goes into proving that the Mullins–Sekerka vector field is well–defined as a map from E^{h} to E^{h−3}. First the case in which there is no boundary is studied. The terms resulting from the Neumann condition when an exterior boundary is introduced are then shown to correspond to an operator which is of lower order, so that MR is not affected. Use is made here of the perturbation result from 1.4.1 as well as the interpolation property 2.2.2. The case of several loops is treated in a similar way. The Dirichlet to Neumann map is decomposed into the sum of the diagonal operator which computes the Dirichlet to Neumann map for the individual loops ignoring interactions between different loops, and a lower order operator.

3.1 Problem formulation

In this section we formulate the Mullins–Sekerka interface motion problem as an initial value problem for a concrete evolution equation relative to the Banach space pair (E^{h−3}, E^{h}). That is to say, we determine the vector field F⃗ : E^{h} → E^{h−3} such that the Mullins–Sekerka problem is equivalent to the initial value problem:

\begin{aligned}ḋ(t) & = F⃗ (d(t)) \\ d(0) & = d_{0} \end{aligned}  (3.1)

for a given initial state d_{0} ∈ E^{h}. As the state variable d we choose the parametrization of the interface curve in terms of polar coordinates (r, θ), where r ∈ ℝ^{m} and θ ∈ [−π, π), relative to predefined points of origin, one for each of the m connected components of the curve. To begin, we discuss the parametrization in more detail. We then compute the normal velocity and curvature of the interface curve in the chosen state space coordinate d. Finally, we write the evolution equation for d.

Parametrization

Let Γ be a connected closed curve in the plane, assumed to be diffeomorphic to the disjoint union of m circles. We also assume each connected component Γ_{j} of Γ to be starshaped about some point O_{j}. In this subsection we parametrize Γ in such a way that it is represented by a function d : S^{1} → ℝ^{m}, where S^{1} is the circle of radius 1, in the form:

r = r_{0} + d(θ)  (3.2)

Here, r ∈ ℝ^{m} will be the m–vector of radial distances to the points O_{j}, while θ ∈ [−π, π) will represent a polar angle (see below). We therefore actually have d : [−π, π) → ℝ^{m}; we make the standard identification of [−π, π) with S^{1}, in which the point θ ∈ [−π, π) is identified with the point e^{iθ} ∈ S^{1}.

We now define the coordinates (r, θ). Choose a reference point O_{j} in the domain enclosed by Γ_{j} in such a way that Γ_{j} is starshaped about O_{j}; also choose a point U_{j} on Γ_{j}, and an orientation for Γ_{j}. The latter is equivalent to choosing a unit vector v⃗_{j} in the tangent space T_{U_{j}}Γ_{j}. Associate to an arbitrary collection of points p = {p_{1}, .., p_{m}|p_{i} ∈ Γ_{j}} the m–vector r(p) of polar radii defined as follows:

r(p) = (r_{j}(p))_{j=1,..,m},  (3.3)

where

r_{j}(p) = the Euclidean distance between p_{j} and O_{j}

The polar angles θ_{j}(p) are given by:

θ_{j}(p) + π = the Euclidean angle between the rays O⃗_{j}U_{j} and O⃗_{j}p_{j}(3.4)

Here, it is understood that the angle θ_{j}(p) increases in the direction of the unit vector v_{j}. Notice that the reference point U_{j} on Γ_{j} corresponds to the value θ_{j} = −π.

We will be interested in curves Γ parametrized in terms of a single scalar θ ∈ [−π, π) as follows:

\begin{aligned}r_{j} & = r_{0,j} + d_{j}(θ_{j}, t), \\ θ_{j} & = θ, \end{aligned}  (3.5)

We will usually write this as:

r = r_{0} + d(θ, t),  (3.6)

For fixed t and θ, the parameters r, r_{0}, and d are in ℝ^{m}. Notice that for each fixed t, the function d(·, t) maps the interval [−π, π) (or, equivalently, the unit circle S^{1}) to a subset of Euclidean m–space ℝ^{m}.

For the sake of notational simplicity, in the remainder of the present section we will restrict our attention to connected curves Γ, i.e. m = 1. We will let r_{0} denote the “base radius” which according to the above conventions would be denoted by (r_{0})_{1}, the first (and only) component of the (m = 1)–vector r_{0}. We will assume d ∈ C(I, E^{h}) ∩ C^{1}(I, E^{h−3}) for some interval I. Here, we think of d as the function t ↦ d(·, t). The analogous statements for curves with several components are straightforward. For example, in general we will have d ∈ C(I, (E^{h})^{m}); in other words, d(θ, t) = (d_{j}(θ, t))_{j=1,..,m}, with each d_{j} ∈ C(I, E^{h}). A few words will be said about this at the beginning of section 3.3.

It will be convenient to use the complex parametrization for Γ:

x(θ, t) = r(θ, t)e^{iθ} = (r_{0} + d(θ, t))e^{iθ}  (3.7)

We will use the notations f(x(θ)) and f(θ) interchangeably when dealing with functions f in some space E^{a} or even in some space C(I, E^{a}), dropping the explicit reference to t for simplicity.

We let s denote the arc length parameter of the curve Γ:

s(p) = the distance between U and p measured along Γ  (3.8)

We assume that the arc length increases in the direction of the vector v. We have the relation

\frac{ds}{dθ} = \sqrt{r^{2} + (\frac{dr}{dθ} )^{2}}  (3.9)

In particular, for a circle of radius r_{0} we have:

\frac{ds}{dθ}|_{r≡r_{0}} = r_{0}  (3.10)

Polar angle vs. arclength parametrization

The functions which appear in our treatment of potential theory below will normally be parametrized in terms of the polar angle θ as explained in the preceding subsection. However, some basic tools which will be used, such as the divergence theorem and certain potential-theoretic results, involve path integrals with respect to arc length. In order to allow us to change from θ to s freely, given a function f(θ) of the polar angle we define a function f̃(s) of arc length in such a way that the associated measures on Γ coincide, i.e. we should have:

f(θ(x))dθ(x) = f̃(s(x))ds(x)  (3.11)

Thus, we simply let

f̃(s) = \frac{f(θ)}{ds/dθ}  (3.12)

The following notational convention should be kept in mind. Even if we are dealing with the polar angle version f(θ), we will sometimes interpret f as a measure on the interface curve Γ as suggested above. Thus, we may write

∫_{Γ} f(x)dθ(x)

instead of

∫_{S^{1}} f(θ)dθ

or

∫_{−π}^{π} f(θ)dθ

Normal velocity

We assume that we are given a time–dependent family of connected curves Γ_{t}, parametrized in polar coordinates as described above, in the form (r = r_{0} + d(θ, t), θ), with d ∈ C(I, E^{h}) ∩ C^{1}(I, E^{h−3}). We compute the normal velocity of Γ at time t as follows:

V_{n} = V⃗ · n̂,  (3.13)

where V⃗ denotes the velocity vector of Γ, the dot · denotes the inner product in the Euclidean plane, and n̂ denotes the outward pointing unit normal vector for Γ at the point of interest. We have the complex parametrization for Γ:

x(θ, t) = r(θ, t)e^{iθ}  (3.14)

from which we find the velocity vector

V⃗ = \frac{∂x}{∂t} = \frac{∂d}{∂t}e^{iθ}  (3.15)

The complex number e^{iθ} represents a unit vector in the radial direction; we denote this vector by r̂. Observe that r̂ · n̂ = θ̂ · T̂, where θ̂ is a positively oriented unit vector in the direction normal to r̂ and T̂ is the positively oriented unit tangent vector to Γ. Thus, the normal velocity of Γ equals

V_{n} = \frac{∂d}{∂t}θ̂ · T̂  (3.16)

The unit vector θ̂ equals ie^{iθ}, and the unit tangent vector T̂ is given by the complex number

T̂ = \frac{\frac{∂x}{∂θ}}{|\frac{∂x}{∂θ}|} = \frac{d_{θ}e^{iθ} + i(r_{0} + d)e^{iθ}}{\sqrt{(r_{0} + d)^{2} + d_{θ}^{2}}}  (3.17)

This yields for the normal velocity

V_{n} = \frac{∂d}{∂t}ℜ ((\frac{d_{θ}e^{iθ} + i(r_{0} + d)e^{iθ}}{\sqrt{(r_{0} + d)^{2} + d_{θ}^{2}}} ) (−i)e^{−iθ}) = \frac{d_{t}(r_{0} + d)}{\sqrt{(r_{0} + d)^{2} + d_{θ}^{2}}} (3.18)

Since clearly d_{t} ∈ C(I, E^{h−3}) and d_{θ} ∈ C(I, E^{h−1}), the results in the preceding chapter allow us to infer that the normal velocity V_{n} is in the space C(I, E^{h−3}).

Curvature

We assume there is given a curve Γ parametrized in polar coordinates as

x(θ) = re^{iθ},

where

r = r_{0} + d(θ)

The time-dependence of d will be supressed for the sake of notational simplicity. We will compute the curvature of Γ, which we denote by κ(d). We have the formula

κ(d) = \frac{\frac{∂x}{∂θ} × \frac{∂^{2}x}{∂θ^{2}}}{|\frac{∂x}{∂θ}|^{3}}  (3.19)

The vertical bars denote the Euclidean norm of the enclosed vector. Below, we use primes to denote differentiation with respect to the polar angle θ. We have:

\frac{∂x}{∂θ} = (d′ + i(r_{0} + d))e^{iθ}  (3.20)

and

\frac{∂^{2}x}{∂θ^{2}} = (d′′ − (r_{0} + d) + 2id′)e^{iθ}  (3.21)

The curvature equals

\begin{aligned}κ(d) & = \frac{−ℑ (((d′ + i(r_{0} + d))e^{iθ}) (d′′ − (r_{0} + d) − 2id′)e^{−iθ})}{((r_{0} + d)^{2} + d′^{2})^{3/2}} \\ & = \frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}} \end{aligned} (3.22)

We see using Theorems 2.3.1 and 2.3.3 that κ(d) ∈ E^{h−2} if d ∈ E^{h} with sufficiently small E^{h} norm. By Lemma 2.5.2, the mapping d ↦ κ(d) is analytic from a neighborhood of 0 in E^{h} to E^{h−2}. If we have a time–dependent interface d ∈ C^{1}(I, E^{h−3}) ∩ C(I, E^{h}), then we will have κ(d) ∈ C(I, E^{h−2}).

The equation of motion

We may now write the Mullins–Sekerka equation of motion. The state variable is the interface parametrization d. We parametrize the interface Γ in polar coordinates (r, θ), in the form

r = r_{0} + d(θ, t).

The state pair is (X, Y) = (E^{h−3}, E^{h}). We assume d ∈ C(I, E^{h}) ∩ C^{1}(I, E^{h−3}). Using the definition of the Mullins–Sekerka problem in section 1.1, and substituting for the normal velocity its expression in terms of d as found in 3.18, we see that the equation of motion for d is:

d_{t} \frac{(r_{0} + d)}{\sqrt{(r_{0} + d)^{2} + d′^{2}}} = N(d)κ(d),  (3.23)

where N(d) is the two phase Dirichlet to Neumann map for the (domain, curve) pair (Ω, Γ) and κ(d) is the curvature of Γ. Expanding the curvature term κ(d) as in Eq. 3.22, we obtain:

d_{t} = \frac{\sqrt{(r_{0} + d)^{2} + d′^{2}}}{(r_{0} + d)} N(d)\frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}}  (3.24)

The right-hand side of this equation is the Mullins-Sekerka vector field, which we will denote by F⃗ (d).

Linearization

The proof of maximal regularity requires information about the linearization at the initial state d = 0 of the vector field for the Mullins–Sekerka problem. As found above in 3.24, three terms are involved:

\frac{\sqrt{(r_{0} + d)^{2} + d′^{2}}}{(r_{0} + d)} , N(d), \frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}} (3.25)

We will obtain the linearization of the Dirichlet-to-Neumann map N(d) in the next section. We give the linearizations of the other terms here. For the first term, we find:

δ (\frac{\sqrt{(r_{0} + d)^{2} + d′^{2}}}{(r_{0} + d)} )_{d=0} = 0  (3.26)

For the other term, the curvature operator κ(d), a simple calculation shows:

κ′(0)d = \frac{−d′′ − d}{r_{0}^{2}}  (3.27)

3.2 The case of a single loop

We now address the analyticity of the Dirichlet-to-Neumann map and the analyticity and maximal regularity of the Mullins–Sekerka vector field. For simplicity, in this first section we assume that the interface curve has only one connected component. Moreover, to begin with we ignore the domain surrounding the interface curve, and assume the exterior boundary to be located at infinity. We then treat the more general problem in which a bounded simply connected surrounding domain and the associated Neumann boundary condition are taken into account. The case in which the interface curve has several connected components is dealt with in a separate section.

3.2.1 Analyticity of the Dirichlet to Neumann map

Recall that N(d)f is defined to be the jump of the derivative, perpendicular to the curve Γ parametrized in polar coordinates by r = r_{0} + d(θ), of the harmonic extension to the domain bounded by Γ of the given boundary value f. In other words, we have

N(d)f = (∂_{n}u)|_{Γ^{+}} − (∂_{n}u)|_{Γ^{−}}  (3.28)

where ∂_{n} denotes differentiation in the direction of the outward pointing normal to Γ and u satisfies the Dirichlet boundary-value problem

\begin{aligned}Δu & = 0 in ℝ^{2} \ Γ \\ u & = f on Γ \end{aligned}  (3.29)

We also impose the condition that the flux of u vanish at infinity:

∇u(x) = O(|x|^{−2}) as |x| → ∞  (3.30)

This condition is required for uniqueness in the Dirichlet problem [15]. We will prove the following result.

Theorem. Let Γ be a connected starshaped curve parametrized in polar coordinates by d ∈ E^{h} with h > 4. Then the Dirichlet to Neumann map N(d) maps E^{h−2} continuously into E^{h−3}. N(d) is analytic as a map d ↦ N(d) from E^{h} to C(E^{h−2}, E^{h−3}).

Proof. Let d ∈ E^{h} with h > 4, and let f ∈ E^{h−2}. We seek to find the solution u of the boundary-value problem defined by Eqs. 3.29 and 3.30, and then to extract the jump [∂u/∂n] which appears on the right-hand side of Eq. 3.28. The jump conditions from classical potential theory [15], [19] suggest that we assume u to be represented as the potential of a single layer distributed on the interface curve. In other words, letting σ_{1} ∈ E^{h−3} denote the density of the single layer \underline{relative to the polar angle measure dθ}, the boundary values of u are assumed to be given up to an additive constant by the function S(d)σ_{1} defined as follows:

(S(d)σ_{1})(θ_{1}) = ∫_{−π}^{π} k(d)(θ_{1}, θ_{2})σ_{1}(θ_{2})dθ_{2}  (3.31)

Here, the kernel k(d) is the polar coordinate version of the free space Green function in the Euclidean plane, evaluated on the interface which is parametrized by r = r_{0} + d(θ):

k(d)(θ_{1}, θ_{2}) = \frac{1}{2π} log |x(θ_{1}) − x(θ_{2})|  (3.32)

Above, x(θ) is as in Eq. 3.7. We point out for future reference that since d ∈ E^{h} ⊆ C^{3} if h > 4, and since therefore also x ∈ C^{3} by 3.7, we have:

|x(θ_{1}) − x(θ_{2})|^{2} = O ((θ_{1} − θ_{2})^{2})  (3.33)

as the difference θ_{1} − θ_{2} → 0 (in fact, a direct computation shows that the derivative with respect to θ_{1}, keeping θ_{2} fixed, of the square of the distance |x(θ_{1}) − x(θ_{2})|^{2}, is twice the quantity

r_{θ}(θ_{1})r(θ_{1}) − r_{θ}(θ_{1})r(θ_{2}) cos(θ_{2} − θ_{1}) + r(θ_{2})r(θ_{1}) sin(θ_{2} − θ_{1}),

which equals 0 at θ_{1} = θ_{2}, thus implying that the square of the distance is of order O(|θ_{1} − θ_{2}|^{2}) as θ_{1} − θ_{2} → 0, since x is C^{2}).

The function S(d)σ_{1}, which is defined in Eq. 3.31 only on the interface, extends to a harmonic function v defined at all points x by:

v(x) = \frac{1}{2π} ∫_{Γ} log |x − y|σ_{1}(y) dθ(y)  (3.34)

As we suggested above, an additive constant is needed to complete the specification of the solution u of the boundary-value problem defined by Eqs. 3.29 and 3.30. We let

u(x) = v(x) + c  (3.35)

where v is as in Eq. 3.34 and c is a constant to be determined.

The normal derivative jump; polar vs arclength densities. The jump of the normal derivative of u across the interface is then equal to the single-layer density σ̃_{1} \underline{relative to standard arclength measure ds} [19], [15]. Since the ds and dθ versions of the single-layer density are related by

σ_{1}dθ = σ̃_{1}ds,  (3.36)

we have:

[\frac{∂u}{∂n}] = \frac{σ_{1}}{\frac{ds}{dθ}}  (3.37)

From this point onward, path integrals will be performed with respect to the polar angle variable, and densities will be specified relative to this variable also, unless otherwise stated. The net effect of this convention is the need to divide by ds/dθ in computing the Neumann jump in Eq. 3.37.

Functions with 0 mean and the projection P_{0}. Harmonicity of the potential u at infinity, as required by Eq. 3.30, is equivalent to the requirement that the single-layer density σ_{1} have zero average on Γ (see [15]):

∫_{Γ} σ_{1}(x) dθ(x) = 0  (3.38)

Observe that although the integral in Eq. 3.38 is performed with respect to the polar angle measure and not the usual arclength measure, the integral has the same value as the more standard version in which σ_{1}dθ is replaced by σ̃_{1}ds.

We let P_{0} denote the projection onto the subspace of functions of mean 0 in the space E^{h−3}:

P_{0} = I − \frac{1}{2π} ∫_{Γ}( )dθ  (3.39)

Then Eq. 3.38 becomes:

P_{0}σ_{1} = σ_{1}  (3.40)

On the interface Γ, we enforce the Dirichlet boundary condition, which by Eqs. 3.31, 3.34, and 3.35 becomes:

f = S(d)σ_{1} + c  (3.41)

Here, we assume that f ∈ E^{h−2}, with h > 4. By using the assumption that σ_{1} integrates to 0 it may be shown that, in the case of a circle (d = 0), the term S(0)σ_{1} has integral 0 over Γ. It follows that the constant c is given in this case by:

c = (I − P_{0})f = \frac{1}{2π} ∫_{Γ} f(x) dθ(x)  (3.42)

In any case, the projection P_{0} eliminates the constant c in Eq. 3.41, yielding:

P_{0}f = P_{0}S(d)σ_{1}  (3.43)

To obtain the Dirichlet to Neumann map, we invert this equation, solving for the single-layer density σ_{1} = S^{−1}P_{0}f in terms of the given Dirichlet data Sσ_{1} +c = f on the boundary. The operator S(d) may not be invertible as a map from E^{h−3} to E^{h−2}. However, we show below (see the comments after Eq. 3.54) that P_{0}S(d) is always invertible as a map from the space P_{0}E^{h−3} of E^{h−3} functions with mean 0, to the corresponding subspace P_{0}E^{h−2} of E^{h−2}. Using Eq. 3.37, we may therefore write

N(d) = \frac{1}{ds/dθ}S(d)^{†}P_{0},  (3.44)

where S(d)^{†} is the pseudoinverse of S(d) from P_{0}E^{h−3} to P_{0}E^{h−2} (which is actually a true inverse map as described). We now proceed to study the single-layer integral operator S(d).

The single-layer integral operator for a circle. For the circle of radius r_{0}, d = 0, one finds that the integral kernel in Eq. 3.32 above equals

k(0)(θ_{1}, θ_{2}) = \frac{1}{4π} log (4r_{0}^{2} sin^{2} (\frac{θ_{1} − θ_{2}}{2} ))

= \frac{1}{4π} log (4 sin^{2} (\frac{θ_{1} − θ_{2}}{2} )) + \frac{1}{2π} log r_{0}

(3.45)

Since the function \frac{1}{2} log(sin^{2} \frac{θ}{2}) is the integral of the function \frac{1}{2 tan \frac{θ}{2}} which appears in Eq. 2.59, and whose Fourier coefficients are −\frac{i}{2}sgnk, it follows that S(0) is a convolution operator with Fourier symbol \frac{−sgnk}{2k} + \frac{1}{2π}(log r_{0})δ_{k,0}. In terms of the periodic Hilbert transform H, which is defined by convolution with \frac{1}{2 tan \frac{θ}{2}} and therefore has the symbol −\frac{i}{2} sgnk, we have

S(0) = \frac{1}{2π} HD^{−1}P_{0} + \frac{1}{2π} (log r_{0})(I − P_{0})  (3.46)

Here, D denotes differentiation in the θ variable, and the integration constant in D^{−1} is chosen in such a way that D^{−1}ϕ integrates to 0 on Γ; thus, we have D^{−1} = P_{0}D^{−1}. Also, H = P_{0}H. Notice that we are using the symbol P_{0} to denote the projection onto the subspace of functions of mean zero in several different ambient spaces. For example, the operator P_{0} in the term P_{0}D^{−1} is defined on the space E^{h−2}, while the operator I −P_{0} is defined on E^{h−3}. Observe that the range of the latter operator consists of constant functions, and therefore is contained in E^{∞}.

We assume that lengths have been scaled in such a way that r_{0} < 1, so that for any a, the operator S(0) maps the space E^{a} isomorphically to the space E^{a+1}. Observing that (isgnk)^{−1} = −isgnk, the Dirichlet to Neumann map in the case of a circle may now be written as:

N(0) = \frac{1}{r_{0}} (\frac{1}{2π} HD^{−1} + \frac{1}{2π} (log r_{0})(I − P_{0}))^{−1} P_{0}

= \frac{1}{r_{0}} (P_{0} \frac{1}{2π} HD^{−1}P_{0} + (I − P_{0})\frac{1}{2π} (log r_{0})(I − P_{0}))^{−1} P_{0}

= \frac{1}{r_{0}} (−D2πHP_{0} + 2π(log r_{0})^{−1}(I − P_{0})) P_{0}

= −\frac{2π}{r_{0}} DHP_{0}

= −\frac{2π}{r_{0}} DH

(3.47)

The single-layer integral operator for a noncircular curve. For general d, the kernel k(d) from Eq. 3.32 becomes:

k(d)(θ_{1}, θ_{2}) = \frac{1}{2π} log |x(θ_{1}) − x(θ_{2})|

= \frac{1}{4π} log (4r_{0}^{2} sin^{2} (\frac{θ_{1} − θ_{2}}{2} )) + \frac{1}{4π} log (\frac{|x(θ_{1}) − x(θ_{2})|^{2}}{4r_{0}^{2} sin^{2} (\frac{θ_{1}−θ_{2}}{2} )} )

(3.48)

We recognize the first term on the right as the kernel for a circle, k(0)(θ_{1}, θ_{2}), from Eq. 3.45. Using Eq. 3.46, we see that the single-layer integral operator S(d) therefore decomposes as follows:

S(d) = \frac{1}{2π} HD^{−1}P_{0} + \frac{1}{2π} (log r_{0})(I − P_{0}) + B(d)  (3.49)

where the operator B(d) satisfies

(B(d)σ_{1}) (x) = \frac{1}{4π} ∫_{−π}^{π} log (\frac{|x(θ_{1}) − x(θ_{2})|^{2}}{4r_{0}^{2} sin^{2} (\frac{θ_{1}−θ_{2}}{2} )} ) σ_{1}(x(θ_{2}))dθ_{2}  (3.50)

B(d) is a continuous map from E^{h−3} to E^{h−2} and depends analytically on d by the results from sections 2.3 and 2.5 in the preceding chapter. We observe that since the numerator in the kernel of B(d) is of order O((θ_{1}−θ_{2})^{2}) by Eq. 3.33, and since the denominator is of order O((θ_{1} − θ_{2})^{2}) and furthermore is independent of d and differs for different values of θ_{2} only by a translation in θ, it follows from Theorem 2.4.1 on analyticity of O(θ^{2}) division that the quotient in the kernel of B(d) is analytic in d as a map from E^{h} to E^{h−2}, and that the region of convergence of the power series expansion of the quotient in d near d = 0 is uniform in θ_{2}. Furthermore, since the numerator in the kernel of B(d) is of order O((θ_{1} − θ_{2})^{2}) for all d ∈ E^{h}, the proof of Theorem 2.4.1 shows that the quotient is globally analytic in the numerator, and therefore in d. This property is preserved by composition with the logarithm function, by Theorem 2.3.3 on analyticity of composition. Lemma 2.5.2 therefore shows that B(d) is analytic in d as a map d ↦ B(d) from E^{h} to C(E^{h−3}, E^{h−2}). By Eq. 3.49, we conclude that the single-layer integral operator S(d) is analytic as a map d ↦ B(d) from E^{h} to C(E^{h−3}, E^{h−2}). For future reference, we also observe that Lemma 2.5.3 shows that B(d) is compact as a map from L^{1} to E^{h−2}.

The Dirichlet-to-Neumann map. The Dirichlet–to–Neumann map N(d) may now be computed from the single-layer integral operator S(d) as follows. Averaging both sides of eq. 3.41 over Γ, we have

c = (I − P_{0})(f − S(d)σ_{1})  (3.51)

and so Eq. 3.41 becomes

P_{0}f = P_{0}S(d)σ_{1}  (3.52)

which yields

σ_{1} = (P_{0}S(d))^{−1}P_{0}f  (3.53)

The invertibility for small d of P_{0}S(d) on the space P_{0}E^{h−2} consisting of E^{h−2} functions with mean zero is clear from the following decomposition (see Eq. 3.49):

P_{0}S(d)P_{0} = P_{0}(S(0) + B(d))P_{0} = \frac{1}{2π} HD^{−1} + P_{0}B(d)P_{0}  (3.54)

since HD^{−1} is invertible as a map from P_{0}E^{h−3} to P_{0}E^{h−2}, B(d) has small norm as a map from E^{h−3} to E^{h−2}, and P_{0} has norm 1. Moreover, it follows from Fredholm–Riesz–Schauder theory that the operator P_{0}S(d) is invertible (as a map from P_{0}E^{h−3} to P_{0}E^{h−2}) for any d for which it is defined. Indeed, B(d) is compact as a map from E^{h−3} to E^{h−2} by Lemma 2.5.3; to show invertibility of P_{0}S(d), it therefore suffices to show injectivity, as this implies injectivity of the adjoint operator and therefore surjectivity of P_{0}S(d). Assume ϕ ∈ P_{0}E^{h−3} is such that P_{0}S(d)ϕ = 0. Then the single-layer potential v produced by the density ϕ according to Eq. 3.34 is constant on Γ. By uniqueness for the Dirichlet problem with the auxiliary harmonicity condition at infinity, v is constant in ℝ^{2}. Thus the Neumann jump ϕ/\frac{ds}{dθ} vanishes, so ϕ vanishes, and injectivity is proved.

The Dirichlet to Neumann map is therefore given by (the dagger † denotes pseudoinverses relative to the appropriate subspaces consisting of functions with mean 0):

\begin{aligned}N(d) & = \frac{1}{ds/dθ}(P_{0}S(d))^{†} \\ & = \frac{1}{\sqrt{(r_{0} + d)^{2} + r′^{2}}} (\frac{1}{2π} HD^{−1} + P_{0}B(d))^{†} \\ & = \frac{1}{\sqrt{(r_{0} + d)^{2} + r′^{2}}} (\frac{1}{2π} HD^{−1} (I − 2πDHP_{0}B(d)))^{†} \\ & = −\frac{2π}{\sqrt{(r_{0} + d)^{2} + r′^{2}}} (I − 2πDHP_{0}B(d))^{−1} DHP_{0} \\ & = −\frac{2π}{\sqrt{(r_{0} + d)^{2} + r′^{2}}} (I − 2πDHB(d))^{−1} DH \end{aligned}  (3.55)

Observe that the range of N(d) as defined here contains only functions with mean 0 on Γ. This guarantees harmonicity at infinity of the associated singlelayer potentials. We proved above that B(d) is analytic as a map d ↦ B(d) from E^{h} to C(E^{h−3}, E^{h−2}). It follows that the operator N(d) is an analytic map d ↦ N(d) from E^{h} to C(E^{h−2}, E^{h−3}) as claimed. This proves Theorem 3.2.1.

□

Linearization of the operator B(d). The subsequent analysis of maximal regularity in the case in which the interface has several connected components will require information concerning the linearization in d of the Dirichlet-to-Neumann map N(d) at d = 0. It is clear from Eq. 3.55 that the only nontrivial term in the linearization arises from B(d).

By using the polar coordinate parametrizations

\begin{aligned}x(θ_{1}) & = (r_{0} + d(θ_{1}))e^{iθ_{1}} \\ x(θ_{2}) & = (r_{0} + d(θ_{2}))e^{iθ_{2}}, \end{aligned}  (3.56)

we find that the kernel of the integral operator B(d) defined in Eq. 3.50 equals

log (\frac{(r_{0} + d(θ_{1}))^{2} + (r_{0} + d(θ_{2}))^{2} − 2(r_{0} + d(θ_{1}))(r_{0} + d(θ_{2})) cos(θ_{1} − θ_{2})}{4r_{0}^{2} sin^{2} (\frac{θ_{1}−θ_{2}}{2} )})

(3.57)

Linearizing the kernel in d at d = 0, we obtain:

\frac{r_{0}(d(θ_{1}) + d(θ_{2}))(1 − cos(θ_{1} − θ_{2}))}{r_{0}^{2}2 sin^{2} (\frac{θ_{1}−θ_{2}}{2} )},  (3.58)

which after cancellation is seen to equal simply

\frac{d(θ_{1}) + d(θ_{2})}{r_{0}}.  (3.59)

Thus, the linearization of B(d) itself is given by:

((B′(0)d)σ_{1})(θ_{1}) = \frac{1}{4πr_{0}} ∫_{−π}^{π}(d(θ_{1}) + d(θ_{2}))σ_{1}(θ_{2})dθ_{2}  (3.60)

From the above expression and the results on integral operators from the preceding chapter, specifically Lemmas 2.5.1 and 2.5.2, it follows that for each d ∈ E^{h} the operator B′(0)d maps L^{1} continuously into E^{h}, and that the map d ↦ B′(0)d is analytic from E^{h} to C(L^{1}, E^{h}).

3.2.2 Maximal regularity for the Mullins–Sekerka vector field

Since the curvature κ(d) maps E^{h} to E^{h−2} analytically for small d, the above result concerning the Dirichlet to Neumann map N(d) implies that the Mullins– Sekerka vector field is analytic as a map from a neighborhood of the origin in E^{h}, to E^{h−3}.

Proving the maximal regularity estimate requires analyzing the linearization of the Mullins–Sekerka vector field F⃗ (d) in d at d = 0. We must compute the following operator:

δ (\frac{\sqrt{(r_{0} + d)^{2} + d′^{2}}}{(r_{0} + d)} N(d)\frac{−d′′(r_{0} + d) + 2d′^{2} + (r_{0} + d)^{2}}{((r_{0} + d)^{2} + d′^{2})^{3/2}} )_{d=0}

Using the fact that the leftmost term linearizes to 0 at d = 0, the linearization of the Mullins-Sekerka vector field reduces to:

δ (F⃗ (d))_{d=0} = N(0)(κ′(0)d) + (N′(0)d)κ(0)  (3.61)

We found N(0) and κ′(0) above in Eq. 3.47 and, respectively, Eq. 3.27, so we can deal with the first term on the right. To find N′(0) := δN_{d=0}, recall from Eq. 3.55 that:

N(d) = −\frac{2π}{\sqrt{(r_{0} + d)^{2} + d′^{2}}} (I − 2πDHB(d))^{−1} DH

Since κ(0) is constant, and since we see that N(d) annihilates constants, we obtain

(N′(0)d)κ(0) = 0

The only term left in the linearization is therefore

δ (F⃗ (d))_{d=0} = N(0)(κ′(0)d) = −\frac{2π}{r_{0}}DHκ′(0)d

The linearized curvature operator was given in Eq. 3.27:

κ′(0)d = \frac{−d′′ − d}{r_{0}^{2}}

It follows that the linearized vector field is given by:

δ (F⃗ (d))_{d=0} = \frac{2π}{r_{0}^{3}} DH(d′′ + d)  (3.62)

The Fourier symbol of the linearized vector field is

−\frac{π}{r_{0}^{3}} |k|(k^{2} − 1)d̂_{k},  (3.63)

It follows immediately that the linearized vector field defines a sectorial operator on the state space E^{h−3}. Also, since the symbol is of order |k|^{3}|d̂_{k}|, it is not difficult to show that the linearized Mullins–Sekerka vector field satisfies the maximal regularity property relative to the state space pair (E^{h−3}, E^{h}). Theorem 3.2.1. Suppose f ∈ C(I, E^{h−3}). Then the function Rf defined below is in C(I, E^{h}), where L is the linearization L = δ (N(d)κ(d))_{d=0} of the Mullins– Sekerka vector field around a circle.

Rf(t) = ∫_{0}^{t} e^{(t−s)L}f(s)ds ∀t ∈ I  (3.64)

Furthermore, the mapping R so defined is continuous from C(I, E^{h−3}) to C(I, E^{h}).

Proof. Taking Fourier transforms in the variation of constants formula appearing in the statement, we obtain

R̂f_{k}(t) = ∫_{0}^{t} e^{(t−s)L̂_{k}} f̂_{k}(s)ds ∀t ∈ I  (3.65)

By the above arguments we know that, modulo a nonzero multiplicative constant, the symbol L̂_{k} is given by

L̂_{k} = −(|k|^{3} − |k|)  (3.66)

We have the bound:

|R̂f_{k}(t)| ≤ ∫_{0}^{t} e^{−(t−s)(|k|^{3}−|k|)}|f̂_{k}(s)|ds ∀t ∈ I  (3.67)

The assumption that f ∈ C(I, E^{h−3}) implies that we have the following bound uniformly in s ∈ I:

|f̂_{k}(s)| ≤ C|k|^{−(h−3)},  (3.68)

where C is given by:

C = ∥f∥_{C(I,E^{h−3})}  (3.69)

We thus obtain

|R̂f_{k}(t)| ≤ C ∫_{0}^{t} e^{−(t−s)(|k|^{3}−|k|)}|k|^{−(h−3)}ds ∀t ∈ I  (3.70)

Integrating, and observing that for |k| ≥ 2 the quotient |k|^{3}/(|k|^{3} − |k|) is at most 4/3, we find that for all t ∈ I,

|R̂f_{k}(t)| ≤ \frac{4}{3}C|k|^{−h} if |k| ≥ 2  (3.71)

In the remaining cases k = 0 and |k| = 1 we have the estimate:

|R̂f_{k}(t)| ≤ ∫_{0}^{t} |f̂_{k}(s)|ds ≤ CT |k|^{−h} ∀t ∈ I,  (3.72)

where T is the length of I. For simplicity, we assume T < \frac{4}{3}. Then from Eqs. 3.71 and 3.72 we have:

|R̂f_{k}(t)| ≤ \frac{4}{3}C| max(|k|, 1)|^{−h} ≤ 2C| max(|k|, 1)|^{−h} ∀t ∈ I, ∀k ∈ ℤ(3.73)

Fix t ∈ I. To prove that (Rf)(t) ∈ E^{h}, we need:

|R̂f_{k}(t)| = o (|k|^{−h}) as |k| → ∞  (3.74)

Let ϵ > 0. Choose ϕ ∈ C(I, E^{∞}) such that

∥f − ϕ∥_{C(I,E^{h−3})} < \frac{ϵ}{4}  (3.75)

(That this is possible follows from an argument similar to that used below to prove that (Rf)(t) depends continuously on t in the E^{h} norm.) By Eq. 3.73 applied to f − ϕ instead of f,

|R̂f_{k}(t) − R̂ϕ_{k}(t)| ≤ \frac{ϵ}{2}|k|^{−h}  (3.76)

Also, by applying the computation leading to Eq. 3.73 to ϕ with varying values of the smoothness index h, we obtain:

R̂ϕ_{k}(t) ∈ E^{∞}  (3.77)

Choose k_{0} ∈ ℕ such that

|R̂ϕ_{k}(t)| ≤ \frac{ϵ}{2}|k|^{−h} ∀|k| ≥ k_{0}  (3.78)

Using Eq. 3.76, we now see that

|R̂f_{k}(t)| ≤ ϵ|k|^{−h} ∀|k| ≥ k_{0}  (3.79)

This proves that 3.74 holds as desired, so that (Rf)(t) ∈ E^{h} for each t ∈ I.

We must now prove that Rf ∈ C(I, E^{h}), i.e. that the dependence of (Rf)(t) on t is continuous in the E^{h} norm. Let t, t′ ∈ I, with t < t′. The difference R̂f_{k}(t′) − R̂f_{k}(t) equals:

∫_{0}^{t′} e^{−(t′−s)(|k|^{3}−k)}f̂_{k}(s)ds − ∫_{0}^{t} e^{−(t−s)(|k|^{3}−k)}f̂_{k}(s)ds =  (3.80)

∫_{t}^{t′} e^{−(t′−s)(|k|^{3}−k)}f̂_{k}(s)ds + (e^{−(t′−t)(|k|^{3}−k)} − 1) ∫_{0}^{t} e^{−(t−s)(|k|^{3}−k)}f̂_{k}(s)ds The first term on the right-hand side in Eq. 3.80 is seen to be O(|t′ − t|)|k|^{−h} by an argument analogous to that used above to obtain Eqs. 3.71 and 3.72. We find

\begin{aligned}|∫_{t}^{t′} e^{−(t′−s)(|k|^{3}−k)}f̂_{k}(s)ds| & ≤ \frac{4}{3}C (1 − e^{−(t′−t)}) |k|^{−h} \\ & ≤ \frac{4}{3}C(t′ − t)|k|^{−h} if |k| ≥ 2 \end{aligned}  (3.81)

and

|∫_{t}^{t′} e^{−(t′−s)(|k|^{3}−k)}f̂_{k}(s)ds| ≤ C(t′ − t)|k|^{−h} if |k| = 0, 1  (3.82)

Therefore, the first term on the right-hand side of Eq. 3.80 satisfies:

|∫_{t}^{t′} e^{−(t′−s)(|k|^{3}−k)}f̂_{k}(s)ds| ≤ 2C(t′ − t)|k|^{−h} ∀k ∈ ℤ  (3.83)

For the second term on the right-hand side of Eq. 3.80, argue as follows. Let ϵ > 0. Since f ∈ C(I, E^{h−3}) and I is compact, there exists δ > 0 such that

∥f(t_{1}) − f(t_{2})∥_{E^{h−3}} < \frac{ϵ}{8} ∀t_{1}, t_{2} ∈ I with |t_{1} − t_{2}| < δ  (3.84)

Choose a finite collection of points t_{1}, ..., t_{N} ∈ I whose \frac{δ}{2}–neighborhoods cover I. For each i ∈ \{1, ..., N\}, choose k_{i} ∈ ℕ such that |f̂_{k}(t_{i})| < \frac{ϵ}{8}|k|^{−(h−3)} whenever |k| ≥ k_{i}. Let k_{0} = max_{i=1,..,N} k_{i}. By Eq. 3.84, we now have

|f̂_{k}(s)| < \frac{ϵ}{4}|k|^{−(h−3)} ∀|k| ≥ k_{0}, ∀s ∈ I  (3.85)

By integrating and using Eq. 3.73 we find that for all |k| ≥ k_{0} we have:

|(e^{−(t′−t)(|k|^{3}−k)} − 1) ∫_{0}^{t} e^{−(t−s)(|k|^{3}−k)}f̂_{k}(s)ds| ≤
∫_{0}^{t} e^{−(t−s)(|k|^{3}−k)} |f̂_{k}(s)| ds < \frac{ϵ}{2}|k|^{−h}  (3.86)

Now let |t′ − t| be sufficiently small so that

|(e^{−(t′−t)(|k_{0}|^{3}−k_{0})} − 1)| < \frac{ϵ}{4C}

Then for any |k| ≤ k_{0}, we have by Eq. 3.73:

|(e^{−(t′−t)(|k|^{3}−k)} − 1) ∫_{0}^{t} e^{−(t−s)(|k|^{3}−k)}f̂_{k}(s)ds| < \frac{ϵ}{2}|k|^{−h}  (3.87)

By Eqs. 3.86 and 3.87, we conclude that the second term on the right-hand side of Eq. 3.80 is less than ϵ in absolute value if |t − t′| is sufficiently small. Since by Eq. 3.83 this also holds for the first term, we conclude that

∥(Rf)(t) − (Rf)(t′)∥_{E^{h}} → 0 as |t − t′| → 0

Thus, Rf ∈ C(I, E^{h}). Eq. 3.73 yields continuity of the map f ↦ Rf from C(I, E^{h−3}) to C(I, E^{h}). This proves the theorem. □

3.2.3 The case of a bounded domain

We now consider the effect of including a no–flux (Neumann) boundary condition on the exterior boundary ∂Ω of the bounded simply connected domain Ω enclosing the interface curve Γ. The Dirichlet to Neumann map N(d) is now defined as follows:

N(d)f = (∂_{n}u)|_{Γ^{+}} − (∂_{n}u)|_{Γ^{−}}  (3.88)

where u solves the boundary-value problem

Δu = 0 in Ω \ Γ
u = f on Γ
\frac{∂u}{∂n} = 0 on ∂Ω^{−}

(3.89)

We independently parametrize the interface curve Γ and the exterior boundary ∂Ω in the manner described in section 3.1. In particular, we assume that the exterior boundary is starshaped about some point O_{∞}, and we consider the associated polar coordinate parametrization:

r = r_{∞} + d_{∞}(θ)  (3.90)

We assume d_{∞} ∈ E^{h} for some h > 4, and f ∈ E^{h−2}.

Analyticity

We will now show that the Dirichlet-to-Neumann map N(d) for the case of a single loop in a bounded domain, as defined by Eqs. 3.88 and 3.89, is analytic as a map d ↦ N(d) from E^{h} to C(E^{h−2}, E^{h−3}).

In order to compute the Dirichlet to Neumann map, we use single-layer potentials as in the preceding subsection. However, we now assume that in addition to the single layer σ_{1} along the interface curve, there is also a single layer σ_{∞} on the exterior boundary. The latter will lead us to consider the adjoint of the double-layer integral operator K on the exterior boundary. We have, letting c denote a constant:

u(x) = ∫_{Γ} k(x, y)σ_{1}(y) dθ(y) + ∫_{∂Ω} k(x, y)σ_{∞}(y) dθ(y) + c  (3.91)

The symbol k denotes the free space Green function for the Laplace operator, as in Eq. 3.32. Although the constant c could be incorporated into the outer boundary layer σ_{∞}, we choose to separate it explicitly in order to simplify the subsequent computation of σ_{∞}.

We will seek σ_{∞} and σ_{1} in E^{h−3}. The primary objective is to find σ_{1}, since the Dirichlet-to-Neumann map is determined by it, through the following jump condition:

[∂u/∂n]_{Γ} = \frac{σ_{1}}{ds/dθ}  (3.92)

The zero flux condition on the exterior boundary will serve to express the other density, σ_{∞}, in terms of σ_{1}.

In equation 3.91, restrict x to lie on the interface Γ and let S_{1,1}(d), respectively S_{1,∞}(d), denote the resulting integral operators, obtained by integrating against the free space Green function along the interface Γ(d), respectively along the exterior boundary ∂Ω:

\begin{aligned}(S_{1,1}(d)σ_{1})(x) & = ∫_{Γ} k(x, y)σ_{1}(y) dθ(y) \\ (S_{1,∞}(d)σ_{∞})(x) & = ∫_{∂Ω} k(x, y)σ_{∞}(y) dθ(y), \end{aligned}  (3.93)

The notation S_{i,j} is meant to suggest that the integration is performed along curve j and the result evaluated on curve i, for i, j ∈ {1, ∞}.

As in subsection 3.2.1, one shows using the O(θ^{2}) division result, Theorem 2.4.1, that S_{1,1}(d) is continuous as a map from E^{h−3} to E^{h−2}, and that the map d ↦ S_{1,1}(d) is analytic from E^{h} to C(E^{h−3}, E^{h−2}). It is simpler to deal with the operator S_{1,∞} because the points x and y in the integral defining S_{1,∞}(d)σ_{∞} lie on disjoint curves, and therefore the denominator in the polar coordinate expression for the kernel of the operator S_{1,∞}(d) does not vanish. One sees that the kernel k(d)(·, θ_{2}) of S_{1,∞}(d) is in E^{h}, and that its E^{h} norm is bounded uniformly in θ_{2}, so that S_{1,∞}(d) maps E^{h−3} continuously into E^{h} by Lemma 2.5.1, and the map d ↦ S_{1,∞}(d) is analytic from E^{h} to C(E^{h−3}, E^{h}) by Lemma 2.5.2.

A simple calculation using the polar coordinate parametrizations

\begin{aligned}x & = r(θ_{1})e^{iθ_{1}} \\ y & = r(θ_{2})e^{iθ_{2}} \end{aligned}  (3.94)

produces

k(x, y) = \frac{1}{4π} log (r_{1}^{2} + r_{2}^{2} − 2r_{1}r_{2} cos(θ_{2} − θ_{1})) ,  (3.95)

where r_{i} abbreviates r(θ_{i}) for i = 1, 2. The above claims concerning the kernel may be verified from this equation.

The Dirichlet condition in Eq. 3.89 becomes:

f = S_{1,1}σ_{1} + S_{1,∞}σ_{∞} + c  (3.96)

Projecting onto the space of functions whose mean on Γ is zero, we have

P_{0}f = P_{0}S_{1,1}σ_{1} + P_{0}S_{1,∞}σ_{∞}  (3.97)

The jump of the normal derivative equals

[\frac{∂u}{∂n}]_{Γ} = \frac{σ_{1}}{ds/dθ} = \frac{1}{\sqrt{(r_{0} + d)^{2} + d′^{2}}} S_{1,1}(d)^{†}P_{0} (f − S_{1,∞}(d)σ_{∞}) (3.98)

The symbol S_{1,1}(d)^{†} indicates here the pseudo–inverse of the map S_{1,1}(d) from P_{0}E^{h−2} to P_{0}E^{h−3}. The no–flux condition in Eq. 3.89 becomes, using Eq. 3.91:

0 = \frac{∂u}{∂n} |_{∂Ω^{−}} = ∫_{Γ} k′(x, y)σ_{1}(y) dθ(y) − \frac{1}{2}σ_{∞}(x) + ∫_{∂Ω} k′(x, y)σ_{∞}(y) dθ(y) (3.99)

Here, k′(x, y) denotes the normal differential with respect to the x variable of the free space Green function k(x, y). Normality is thus relative to the tangent space T_{x}∂Ω. If n(x) denotes the unit exterior normal to T_{x}∂Ω, we have:

k′(x, y) = \frac{⟨x − y | n(x)⟩}{2π|x − y|^{2}}  (3.100)

Since both the interface curve Γ and the exterior boundary ∂Ω are in E^{h} with h > 4 and therefore in C^{3}, the numerator of k′(x, y) here is O(θ^{2}). This may be seen by calculating in the polar coordinate parametrizations given in Eq. 3.94, yielding:

k′(x, y) = \frac{r_{1}^{2} − r_{1}r_{2} cos(θ_{2} − θ_{1}) + r′_{1}r_{2} sin(θ_{2} − θ_{1})}{2π (r_{1}^{2} + r_{2}^{2} − 2r_{1}r_{2} cos(θ_{2} − θ_{1})) (r_{1}^{2} + r′_{1}^{2})^{1/2}}  (3.101)

From this expression for the kernel and from the fact that the polar radii are C^{2} (they are in fact C^{3}, of course) in the polar angles θ_{i}, it is easy to see that if both x and y are restricted to lie in the same connected curve, either Γ or ∂Ω, then the quotient is of the form O(θ^{2})/O(θ^{2}), where θ = θ_{2} − θ_{1}. On the other hand, it is clear that if one of x and y is on Γ and the other is on ∂Ω, then the denominator does not vanish and the quotient is nonsingular. In the first case, Theorem 2.3.1 on products and Theorem 2.4.1 on O(θ^{2}) division show that the kernel k′(d)(·, θ_{2}) is in E^{h−3} uniformly in θ_{2}. In the second case, the kernel is in E^{h−1}.

In operator notation, we may write Eq. 3.99 expressing the vanishing of the flux as

(−\frac{1}{2}I + K_{∞,∞}^{∗}) σ_{∞} = −K_{∞,1}^{∗}(d)σ_{1},  (3.102)

where K_{∞,1}^{∗}(d)σ_{1}(x) and K_{∞,∞}^{∗}σ_{∞}(x) are defined for x ∈ ∂Ω by:

\begin{aligned}(K_{∞,1}^{∗}(d)σ_{1})(x) & = ∫_{Γ} k′(x, y)σ_{1}(y) dθ(y) \\ (K_{∞,∞}^{∗})σ_{∞}(x) & = ∫_{∂Ω} k′(x, y)σ_{∞}(y) dθ(y) \end{aligned}  (3.103)

As above, the notation K_{i,j}^{∗} reflects the fact that in each case the integration is performed along curve j and the result evaluated on curve i, where i, j ∈ {1, ∞}.

Although our notation supresses the dependence on the parametrization d in the boundary operator K_{∞,∞}^{∗}, the parametrization of the exterior boundary does play an important role here, as can be seen in Eq. 3.101 above.

Notice that for y ∈ Γ, the kernel k′(·, y) is in E^{h−1} on ∂Ω relative to the standard polar coordinate parametrization and therefore the operator K_{∞,1}^{∗}(d) maps E^{h−3} into E^{h−1} analytically in d ∈ E^{h}. On the other hand, if y ∈ ∂Ω, then since ∂Ω is in E^{h} with h > 2 and therefore C^{1}, the numerator of the kernel k′(x, y) is O(θ^{2}), so by the O(θ^{2}) lemma 2.4.2 k′(·, y) is in E^{h−3} and thus the operator K_{∞,∞}^{∗} is continuous from E^{h−3} to E^{h−3} analytically in d ∈ E^{h}.

We now wish to solve for σ_{∞} in terms of σ_{1} in Eq. 3.102. By Lemma 2.5.3, K_{∞,∞}^{∗} is compact as a map on E^{h−3}. Therefore, Fredholm–Riesz–Schauder theory applies. The solvability condition for σ_{∞} in E^{h−3} is the same as the L^{2} solvability condition that K_{∞,1}^{∗}σ_{1} be orthogonal in L^{2} to the kernel of the operator

− \frac{1}{2}I + K_{∞,∞},  (3.104)

associated with the exterior Dirichlet problem for double-layer potentials on ∂Ω. This is true for the following reason. First, the right-hand side K_{∞,1}^{∗}σ_{1} of Eq. 3.102 belongs to L^{2} for any σ_{1} ∈ E^{h−3} since K_{∞,1}^{∗} maps L^{1} into E^{h−1} ⊆ L^{2}. Now assume we can find σ_{∞} ∈ L^{2} satisfying Eq. 3.102. Recall that the operator K_{∞,∞}^{∗} which appears in Eq. 3.102 maps L^{1} ⊇ L^{2} into E^{h−3}. Since from Eq. 3.102 we can write σ_{∞} = 2 (K_{∞,∞}^{∗}σ_{∞} + K_{∞,1}^{∗}σ_{1}), we see that σ_{∞} is the sum of two E^{h−3} functions and therefore belongs to E^{h−3}. Thus, E^{h−3} solvability indeed follows from L^{2} solvability.

We now examine the L^{2} solvability for σ_{∞} in Eq. 3.102. We will show that the kernel of the operator in 3.104 consists precisely of the constant functions on ∂Ω. This will imply that the solvability condition is simply:

∫_{∂Ω} K_{∞,1}^{∗}σ_{1}(y)dθ(y) = 0  (3.105)

Let us proceed to verify the claim concerning the kernel N of the operator in Eq. 3.104. First, it is known that N contains all constant functions [15]. It therefore suffices to show that N is one-dimensional. By Fredholm theory, it suffices to show that the adjoint operator

− \frac{1}{2}I + K_{∞,∞}^{∗}  (3.106)

for which we wish to prove solvability in Eq. 3.102, has a one-dimensional kernel. The operator in 3.106 is associated with the interior Neumann problem for singlelayer potentials. Its kernel N^{∗} consists of all single-layer densities β on ∂Ω such that the associated single-layer potential v := S_{·,∞}β satisfies ∂v/∂n = 0 on ∂Ω^{−}. It follows by the divergence theorem and the harmonicity of v that v must be constant inside Ω.

Since we know that the kernel N of the operator in 3.104 has nonzero dimension, the same holds by Fredholm theory for N^{∗}, the kernel of the operator in 3.106. Choose β_{0} ≠ 0 in N^{∗}. First we show (recall that by definition, β_{0} is a density relative to the polar angle measure):

∫_{∂Ω} β_{0}dθ ≠ 0  (3.107)

Assume otherwise. Since β_{0} ∈ N^{∗}, the single-layer potential v := S_{·,∞}β is constant inside Ω. The assumption that β_{0} has 0 mean on ∂Ω implies that the gradient ∇v decays like 1/r^{2} at ∞, where r is the polar radius. By the divergence theorem and the harmonicity of v, it follows that v is constant outside Ω as well, and since single-layer potentials are continuous, we conclude that v must be identically constant in ℝ^{2}. The Neumann jump β_{0} therefore vanishes, contrary to our initial assumption. Thus, Eq. 3.107 must hold. Without loss of generality, assume

∫_{∂Ω} β_{0}dθ = 1  (3.108)

Now choose an arbitrary β ∈ N^{∗}. Then the density β̃ = β − (\frac{1}{2π} ∫_{∂Ω} βdθ) β_{0} has zero mean, and the argument just given shows that β̃ = 0, i.e.

β = (\frac{1}{2π} ∫_{∂Ω} βdθ) β_{0}

In particular, N^{∗} is one-dimensional. We conclude that the kernel N of the operator in Eq. 3.104 is also one-dimensional, and therefore consists precisely of the constant functions as claimed.

We will now show that the resulting solvability condition, which was given in Eq. 3.105, holds automatically. Using the divergence theorem, we see that because of the harmonicity of the potential u, the integral of σ_{1} along Γ must vanish if the zero flux condition on the exterior boundary is to be satisfied (notice that the integrals involving ∂u/∂n here are performed with respect to arclength measure ds):

∫_{Γ} σ_{1}(x) dθ(x) = ∫_{Γ} [\frac{∂u}{∂n}]_{Γ} (x) ds(x)

= ∫_{Γ^{+}} \frac{∂u}{∂n}(x) ds(x) − ∫_{Γ^{−}} \frac{∂u}{∂n}(x) ds(x)

= ∫_{Ω\Int Γ} Δu + ∫_{∂Ω} \frac{∂u}{∂n} − ∫_{Int Γ} Δu

= 0.

(3.109)

This condition is the same as that required for harmonicity at infinity in the preceding section.

A similar calculation shows that the integral of K_{∞,1}^{∗}(d)σ_{1} along the exterior boundary ∂Ω equals the integral of σ_{1} along the interface Γ (in physical language, the flux through ∂Ω equals the enclosed mass, in appropriate units). To see this, let v denote the single-layer potential generated by the single-layer density σ_{1} on Γ, i.e.

v(x) = ∫_{Γ} k(x, y)dθ(y)

Then we have, using the divergence theorem:

∫_{∂Ω} K_{∞,1}^{∗}(d)σ_{1} dθ(x) = ∫_{∂Ω^{−}} \frac{∂v}{∂n}(x) ds(x)

= ∫_{Γ^{+}} \frac{∂v}{∂n}(x) ds(x) + ∫_{Ω\Int Γ} Δv

= ∫_{Γ^{+}} \frac{∂v}{∂n}(x) ds(x)

= ∫_{Γ^{+}} \frac{∂v}{∂n}(x) ds(x) − ∫_{Int Γ} Δv

= ∫_{Γ} [\frac{∂v}{∂n}] (x) ds(x)

= ∫_{Γ} σ_{1}(x) dθ(x)

(3.110)

Thus, if we require σ_{1} to have mean 0 on Γ, as Eq. 3.109 shows we must in order to satisfy the zero flux condition on the exterior boundary ∂Ω, then by Eq. 3.110 the solvability condition in Eq. 3.105 holds, and we can indeed find σ_{∞} satisfying Eq. 3.102. However, the inverse of the map −\frac{1}{2}I + K_{∞,∞}^{∗} does not exist, as the kernel of the operator being inverted consists precisely of all singlelayer densities on the boundary ∂Ω whose associated potential is constant (see [15]). Since we may incorporate all such constant potentials into the constant c in equation 3.91, we will select σ_{∞} to lie in the L^{2}–orthogonal complement of the kernel of −\frac{1}{2}I + K_{∞,∞}^{∗}. This is equivalent to obtaining σ_{∞} from Eq. 3.102 by using the corresponding pseudoinverse:

σ_{∞} = − (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(d)σ_{1}.  (3.111)

This effectively eliminates σ_{∞} and allows us to solve for σ_{1}, and therefore for the normal derivative jump, in Eq. 3.98. We first obtain σ_{1}:

σ_{1} = S_{1,1}(d)^{†}P_{0} (f − S_{1,∞}(d)σ_{∞})  (3.112)

= S_{1,1}(d)^{†}P_{0} (f + S_{1,∞}(d) (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(d)σ_{1})

Solving for the single-layer density σ_{1}, and recalling that by Eq. 3.109 σ_{1} satisfies

P_{0}σ_{1} = σ_{1},  (3.113)

we have

σ_{1} = (I − S_{1,1}^{†}P_{0}S_{1,∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗})^{†} S_{1,1}^{†}P_{0}f,

where the dagger associated with the outer parentheses indicates pseudoinverse relative to the space P_{0}E^{h−3} consisting of functions of mean 0 in E^{h−3}. Since the jump of the normal derivative across the interface curve equals σ_{1} divided by ds/dθ, the Dirichlet to Neumann map is therefore given by

N(d) = \frac{1}{ds/dθ} (I − S_{1,1}^{†}P_{0}S_{1,∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗})^{†} S_{1,1}^{†}
= \frac{1}{ds/dθ} (P_{0}S_{1,1} − P_{0}S_{1,∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗})^{†} P_{0} (3.114)

We claim that the operator appearing on the right-hand side of Eq. 3.114 is invertible as a map from the space P_{0}E^{h−3} to the space P_{0}E^{h−2}. To prove this claim, we use the Fredholm–Riesz–Schauder theory for linear maps of the form identity + compact on a Banach space. This theory is applicable because by Lemma 2.5.3, the operator K_{1,∞}^{∗}(d) is compact as a map from L^{1} to E^{h−1}, and so the map

S_{1,1}(d)^{†}P_{0}S_{1,∞}(d) (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(d)

is compact on E^{h−3}. Thus, invertibility from P_{0}E^{h−3} to P_{0}E^{h−2}, of the operator

P_{0}S_{1,1} − P_{0}S_{1,∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}

which appears on the right-hand side of Eq. 3.114 will follow if we can show that the operator is injective as a map from P_{0}E^{h−3} to P_{0}E^{h−2}, for then the Fredholm–Riesz–Schauder theory shows that its adjoint will also be injective and that the original operator must then be surjective, thus an isomorphism. We proceed to prove injectivity. Suppose ϕ ∈ P_{0}E^{h−3} is such that

(P_{0}S_{1,1}(d) − P_{0}S_{1,∞}(d) (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(d)) ϕ = 0 (3.115)

By Eq. 3.97, this equation means that the harmonic function ξ produced by the superposition of the single-layer potential with density ϕ on the interface Γ and the single-layer potential with density σ_{∞} = (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(d)ϕ on the exterior boundary ∂Ω, produces constant Dirichlet data on Γ. But recall (Eq. 3.102) that the single layer σ_{∞} is constructed to enforce the zero flux condition on the exterior boundary. Thus, ξ is constant on Γ and its normal derivative vanishes on ∂Ω. By subtracting a constant we may assume without loss of generality that ξ vanishes on Γ. By Green’s theorem, in the domain enclosed by Γ we have

∫_{Int Γ} |∇ξ|^{2} = ∫_{Γ} ξ \frac{∂ξ}{∂n} − ∫_{Int Γ} ξΔξ  (3.116)

The first term on the right vanishes because ξ has zero Dirichlet data on Γ, and the second term vanishes because ξ is harmonic in Int Γ. Thus, ξ is constant in Int Γ, and because ξ vanishes on Γ, we see that ξ = 0 in Int Γ. Similarly, but also using the fact that ξ has zero Neumann data on the exterior boundary ∂Ω, we have in the domain lying between Γ and ∂Ω,

∫_{∂Ω\Γ} |∇ξ|^{2} = ∫_{∂Ω} ξ \frac{∂ξ}{∂n} − ∫_{Γ} ξ \frac{∂ξ}{∂n} − ∫_{∂Ω\Γ} ξΔξ = 0  (3.117)

We conclude that if h > 4 (so that E^{h−2} is contained in W^{1,2}), then ξ must vanish. Since ϕ is the jump of ∂ξ/∂n across Γ, ϕ also vanishes. This proves injectivity, and the existence of the inverse map follows from the Fredholm–Riesz–Schauder theory. The results in the previous chapter imply that N(d) maps E^{h−2} to E^{h−3} analytically in d as desired.

Maximal regularity

Next we address the maximal regularity issue. Just as in the boundaryless case, discussed in subsection 3.2.2, in the present case the solution to the boundary value problem 3.89 with constant Dirichlet datum is constant and therefore N(d) annihilates constant functions. It follows that

N′(0)κ(0) = 0  (3.118)

and so the linearization δF⃗ (d)_{d=0} reduces to a single term:

δF⃗ (d)_{d=0} = −N(0) \frac{d′′ + d}{r_{0}^{2}}

To prove maximal regularity, we use a result of Angenent (Lemma 2.5 in [3]) which states that if a continuous linear operator T : Y → X satisfies MR with respect to the pair (X, Y), and if S is such that

∀ϵ ∈ ℝ^{+} ∃C_{ϵ} ∈ ℝ^{+} ∀y ∈ Y ||Sy||_{X} ≤ ϵ||y||_{Y} + C_{ϵ}||y||_{X}, (3.119)

then the perturbed operator T + S also satisfies MR with respect to the phase pair (X, Y). As observed in [3], the property in 3.119 follows immediately if one can show that the operator S is continuous from some interpolation space between X and Y strictly below Y, into X. We will use this fact below. Also, it is obvious that 3.119 will be satisfied if S is continuous as a map from X to X, since then one may even let ϵ = 0. As usual, X = E^{h−3} and Y = E^{h}. The role of the interpolation space will be played by the space E^{4}.

Theorem 3.2.2. The operator

N(0)κ′(0) = \frac{1}{r_{0}} (P_{0}S_{1,1}(0) − P_{0}S_{1,∞}(0) (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(0))^{−1} P_{0}κ′(0)

defined using Eq. 3.114, satisfies MR with respect to the pair (E^{h}, E^{h−3}) if h > 4.

Proof. We will ignore the constant multiplicative factor 1/r_{0} in the Dirichletto-Neumann map N(0), as it clearly does not affect maximal regularity.

For notational simplicity, let

A = P_{0}S_{1,1}(0) : P_{0}E^{h−3} → P_{0}E^{h−2}

and

B = −P_{0}S_{1,∞}(0) (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,1}^{∗}(0) : P_{0}E^{h−3} → P_{0}E^{h}

In fact, B maps L^{1} continuously into E^{h} by the results on integral operators from the end of the previous chapter. More specifically, we know that the kernel k(·, θ_{2}) of the operator S_{1,∞}(0) belongs to E^{h}, with a bound on the E^{h} norm which is uniform in θ_{2}, and similarly that the kernels of K_{∞,1}^{∗}(0) and K_{∞,∞}^{∗} belong to E^{h−1} and E^{h−3} respectively (the latter by the O(θ^{2}) division Theorem 2.4.1), and it follows by Lemma 2.5.1 that B maps L^{1} continuously into E^{h}. Also, we know from Theorem 3.2.1 that A^{−1}κ′(0), the Dirichlet-to-Neumann map in the case in which there is no exterior boundary, satisfies MR relative to the pair (E^{h−3}, E^{h}).

We wish to show that MR holds for the operator (A + B)^{−1}κ′(0), which we view as a perturbation of A^{−1}κ′(0). Write

(A + B)^{−1}κ′(0) − A^{−1}κ′(0) = A^{−1} ((I + BA^{−1})^{−1} − I) κ′(0)
= −A^{−1}(I + BA^{−1})^{−1}BA^{−1}κ′(0), (3.120)

where I denotes the identity map on E^{h−3}. We have for any d ∈ E^{h} and any ϵ > 0:

||(A + B)^{−1}κ′(0) − A^{−1}κ′(0)d||_{E^{h−3}} = || − A^{−1}(I + BA^{−1})^{−1}BA^{−1}κ′(0)d||_{E^{h−3}}

≤ ||A^{−1}||_{E^{h−2}→E^{h−3}}||(I + BA^{−1})^{−1}BA^{−1}κ′(0)d||_{E^{h−2}}
≤ ||A^{−1}||_{E^{h−2}→E^{h−3}}||(I + BA^{−1})^{−1}||_{E^{h−2}→E^{h−2}}||BA^{−1}κ′(0)d||_{E^{h−2}}

≤ ||A^{−1}||_{E^{h−2}→E^{h−3}}||(I + BA^{−1})^{−1}||_{E^{h−2}→E^{h−2}}||B||_{L^{1}→E^{h−2}}||A^{−1}κ′(0)d||_{L^{1}}

≤ C||A^{−1}κ′(0)d||_{E^{1}}
≤ C′||d||_{E^{4}},

The above shows that the perturbation (A + B)^{−1}κ′(0) − A^{−1}κ′(0) is continuous from E^{4} into E^{h−3}. If 4 < h ≤ 7, then we have h − 3 < 4 < h, and the result 2.2.2 from the previous chapter implies that E^{4} is an interpolation space strictly between E^{h−3} and E^{h}, with a logconvex norm estimate. If h > 7, then E^{4} is below E^{h−3} on the E^{a} scale, and so the perturbation is continuous as a map from E^{h−3} to itself. In either case, by the comments preceding the statement of the theorem, the proof is now complete. □

Comments. We point out that the crucial property which allows the preceding proof to go through is that the target operator is of the form (A + B)^{−1}, where A^{−1} is known to satisfy MR relative to (E^{h−3}, E^{h}) and B is smoothing in the sense that it maps L^{1} into E^{h−2} continuously. This argument will be used again in proving that maximal regularity also holds in the case in which the interface has several connected components.

3.3 The case of multiple loops

In this section, the above results concerning analyticity of the Dirichlet to Neumann map and maximal regularity for the Mullins–Sekerka vector field are extended to the case in which the interface has several connected components. The extension requires working in slightly modified function spaces, but the basic ideas are the same as in the single loop case.

We assume the interface has m connected components, each diffeomorphic to the circle S^{1} and starshaped about some point as discussed in section 3.1. Accordingly, we consider the function spaces

E_{m}^{h} = {f : S^{1} → ℝ^{m} | ˆ (f_{j})_{k} = o(|k|^{−h}) as |k| → 0, for each j = 1, .., m}

(3.121)

where f_{j} denotes the j–th component function of f, which may be interpreted as the “restriction of f to the j–th loop”; the various loops are parametrized using the polar angle θ. The norm on E_{m}^{h} is simply the maximum of the standard E^{h} norms on the component loops. Notice that E_{m}^{h} is then linearly isometric to the product of the component E^{h} spaces:

E_{m}^{h} ≅ ∏_{j=1,..,m} E^{h} = (E^{h})^{m}  (3.122)

Below, we will drop the subindex m and use E^{h} to denote the corresponding space of functions on the disjoint union of an arbitrary number of loops.

We define the projection P_{0} : (L^{1})^{m} → (L^{1})^{m} by:

(P_{0}f)_{k} = f_{k} − \frac{1}{2πm} ∑_{j=1,..,m} ∫_{Γ_{j}} f_{j}(x) dθ(x)  (3.123)

The range of P_{0} is the subspace of (L^{1})^{m} consisting of all functions whose integral over the union of the Γ_{k} vanishes. The kernel of P_{0} is the one-dimensional space of functions which are equal to a single constant on all the components of Γ.

The projection P_{0} must be distinguished from the diagonal operator P whose k–th diagonal component is the projection P_{k} : L^{1} → L^{1} for the k–th component Γ_{k} of Γ, defined by:

(P f)_{k} = f_{k} − \frac{1}{2π} ∫_{Γ_{k}} f_{k}(x) dθ(x)  (3.124)

The range of P is the subspace of (L^{1})^{m} consisting of functions whose integral on each component Γ_{k} vanishes. The kernel of P is the m-dimensional space of functions which are constant on each Γ_{k}, without necessarily being equal to the same constant on different components. Thus, while the m conditions P_{k}g_{k} = 0 on the restrictions g_{k} of some function g to the components Γ_{k} guarantee that P_{0}g = 0, the converse is certainly false.

Throughout this section, we assume that the base radii r_{0,j} of the polar coordinate parametrizations of the components of the interface curve Γ (see Eq. 3.5) have been scaled to satisfy:

r_{0,j} < 1  (3.125)

As Eq. 3.49 shows, this assumption guarantees that the associated single-layer integral operators S_{j,j}(d) on the respective loops are invertible for d_{j} = 0, and therefore also for small d_{j} ∈ E^{h}.

3.3.1 Analyticity of the vector field

We let Γ_{1}, ..., Γ_{m} denote the connected components of the interface. We assume that a function f ∈ (E^{h−2})^{m} is given, corresponding to a choice of one E^{h−2} function on each of the component curves Γ_{k}, k = 1, ..., m, and we seek a harmonic extension u of f to Ω − Γ whose flux across the outer boundary ∂Ω vanishes. By this we mean the following. We let f_{k} denote the restriction of f to the curve Γ_{k}, in other words, f_{k} is the k–th component of f. We wish to solve the mixed boundary-value problem

Δu = 0 in Ω \ Γ
u = f_{k} on Γ_{k}, k = 1, ..., m,
\frac{∂u}{∂n} = 0 on ∂Ω^{−},

(3.126)

with a view toward obtaining the Dirichlet-to-Neumann map N(d) defined by:

N(d)f = (∂_{n}u)|_{Γ^{+}} − (∂_{n}u)|_{Γ^{−}}  (3.127)

The function N(d)f should lie in the space (E^{h−3})^{m}, and Eq. 3.127 should be interpreted as a system of m equations, one on each component of the interface curve.

We represent the solution u of the boundary-value problem Eq. 3.126 as the potential associated with a collection of single-layer densities σ_{k}, k = 1, ..., m on the respective curves Γ_{k}, and an additional single-layer density σ_{∞} on the boundary ∂Ω. The convention that densities are specified relative to the polar angle measure dθ rather than arclength measure ds is in place here as it is in the preceding section.

For each x ∈ Ω, we have

u(x) = \frac{1}{2π} ∫_{∂Ω}σ_{∞}(y) log |x − y| dθ(y) + ∑_{k=1,..,m} \frac{1}{2π} ∫_{∂Γ_{k}}σ_{k}(y) log |x − y| dθ(y) + c

(3.128)

We emphasize the fact that the constant c is the same for all components of the interface curve. On the j–th component Γ_{j}, we enforce the Dirichlet condition

f_{j}(x) = S_{j,∞}σ_{∞}(x) + ∑_{k=1,..,m} S_{j,k}σ_{k}(x) + c ∀x ∈ Γ_{j}  (3.129)

Here, the operators S_{j,k} and S_{j,∞} are defined by integration against the free space Green function \frac{1}{2π} log |x − y|, as in Eq. 3.128. Notice that S_{j,k} is obtained by integrating along the curve Γ_{k} and then evaluating on the curve Γ_{j}, for j and k in {1, ..., m, ∞}.

Define the matrix operators S_{∞}, K_{∞}^{∗}, S by:

S_{∞} = [ S_{1,∞} ; ⋮ ; S_{m,∞} ] , K_{∞}^{∗} = [K_{∞,1}^{∗}, …, K_{∞,m}^{∗}] , S = [ \begin{matrix}S_{1,1} & … & S_{1,m} \\ ⋮ & ... & ⋮ \\ S_{m,1} & … & S_{m,m} \end{matrix} ]

By applying P_{0} to Eq. 3.129 we obtain:

P_{0}f = P_{0} (S_{∞}σ_{∞} + Sσ)  (3.130)

The zero flux condition on the exterior boundary ∂Ω requires:

0 = (−\frac{1}{2}I + K_{∞,∞}^{∗}) σ_{∞}(x) + K_{∞}^{∗}σ(x) ∀x ∈ ∂Ω  (3.131)

The harmonicity of the potential u defined by Eq. 3.128 implies that the solvability condition for σ_{∞} in Eq. 3.131 is equivalent to vanishing of the \underline{sum} of the integrals of the various single-layer densities σ_{k}, and the latter condition is itself equivalent to vanishing of the flux of u across the outer boundary ∂Ω:

\begin{aligned}∫_{∂Ω^{−}} K_{∞}^{∗}σ(x) dθ(x) & = ∫_{∂Ω^{−}} ∑_{k=1,..,m} K_{∞,k}^{∗}σ_{k}(x) dθ(x) \\ & = ∑_{k=1,..,m} ∫_{Γ_{k}} σ_{k}(x) dθ(x) \\ & = ∑_{k=1,..,m} ∫_{Γ_{k}} [\frac{∂u}{∂n}]_{Γ_{k}} (x) ds(x) \\ & = ∑_{k=1,..,m} ∫_{Γ_{k}^{+}} \frac{∂u}{∂n}(x) ds(x) \\ & = ∫_{∂Ω^{−}} \frac{∂u}{∂n}(x) ds(x) = 0 \end{aligned}  (3.132)

We assume that the σ_{k} satisfy this condition. In terms of the projection P_{0}, Eq. 3.132 states that we must have:

P_{0}σ = σ  (3.133)

Observe that Eq. 3.132 is the Fredholm solvability condition for σ_{∞} in Eq. 3.131, since the kernel of the adjoint of the operator −\frac{1}{2}I + K_{∞,∞}^{∗} consists precisely of the constant functions on ∂Ω. From Eq. 3.131 we can therefore solve for σ_{∞}:

σ_{∞} = − (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} ∑_{k=1,..,m} K_{∞,k}^{∗}σ_{k}  (3.134)

Notice that we do not claim invertibility of the operator −\frac{1}{2}I + K_{∞,∞}^{∗} here. This operator has a one-dimensional kernel. We choose the pseudoinverse relative to the L^{2}-orthogonal complement of this kernel. Substituting this result into Eq. 3.129, we have

P_{0}f = P_{0} (−S_{∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞}^{∗} + S) σ  (3.135)

This is a system of m equations which we can solve for the m densities σ_{1}, ..., σ_{m}. We assume f_{1}, ..., f_{m} ∈ E^{h−2} and we seek σ_{1}, ..., σ_{m} ∈ E^{h−3}. Since σ_{k}/\frac{ds}{dθ} is the jump of the normal derivative of u across Γ_{k}, we thus obtain the Dirichlet to Neumann map.

Let us proceed. Separate the matrix single-layer integral operator S from Eq. 3.130 into the sum of its diagonal part S_{1}, whose i–th diagonal component equals S_{i,i}, and its nondiagonal part S_{2} := S − S_{1}. The system from Eq. 3.135 becomes:

P_{0}f = P_{0} (S_{1} + B) σ,  (3.136)

where

B = S_{2} − S_{∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞}^{∗}  (3.137)

The Dirichlet-to-Neumann map N(d) is obtained by inverting the system which appears in Eq. 3.136:

N(d) = (S_{1} + B)^{†} P_{0}  (3.138)

The pseudoinverse here is computed on the range of P_{0}. Recall the definitions of the projection matrices P_{0} and P from Eqs. 3.123 and 3.124. Since S_{1}(0) is invertible by 3.125, the preceding equation may also be written in the case d = 0 as:

N(0) = P_{0}S_{1}^{−1}P_{0}

To show solvability in Eq. 3.135 (or, equivalently, in Eq. 3.136), we proceed as in the single loop case. By using the polar coordinate expression for the kernel k′(x, y) from Eq. 3.101, with (r_{1}, θ_{1}) and (r_{2}, θ_{2}) replaced by the appropriate expressions corresponding to the curves Γ_{j} and Γ_{k}, we see that the operator K_{j,k}^{∗} maps E^{h−3} continuously into E^{h−1} if j ≠ k, while each operator K_{k,k}^{∗} maps E^{h−3} continuously into itself. The single-layer integral operators S_{j,k} map E^{h−3} continuously into E^{h}, respectively E^{h−2}, assuming j ≠ k, respectively j = k. This argument involves, as usual, the product Theorem 2.3.1, the composition Theorem 2.3.3, the O(θ^{2}) division Theorem 2.4.1, and Lemma 2.5.1 on continuity of integral operators on the E^{a} spaces.

We obtain compactness of K_{∞,k}^{∗} as a map from E^{h−3} to E^{h−1} by using Lemma 2.5.3. It follows that the operator S_{∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞}^{∗} is compact as a map from (E^{h−3})^{m} to (E^{h−2})^{m}. Similarly, since the operator S_{2} involves the single-layer kernel k(x, y) away from singularities only, we see by Lemma 2.5.3 that S_{2} is compact as a map from (E^{h−3})^{m} to (E^{h−2})^{m}.

In order to use the Fredholm–Riesz–Schauder theory to reduce solvability of the system in Eq. 3.135 to injectivity of the operator (system) appearing on the right-hand side of Eq. 3.135, viewed as a map from P_{0}(E^{h−3})^{m} to P_{0}(E^{h−2})^{m}, we must show that the operator which must be inverted in Eq. 3.135 is of the form invertible + compact from P_{0}(E^{h−3})^{m} to P_{0}(E^{h−2})^{m}. By the comments made in the preceding paragraph, it now suffices to prove that the operator P_{0}S_{1}(0) = P_{0}S_{1}(0) is invertible as a map from P_{0}(E^{h−3})^{m} to P_{0}(E^{h−2})^{m}. Analytic dependence on d ∈ (E^{h})^{m} then guarantees invertibility for sufficiently small d ∈ (E^{h})^{m}.

Given f ∈ (E^{h−2})^{m}, we wish to solve the following equation for σ ∈ (E^{h−3})^{m}:

P_{0}f = S_{1}σ,  (3.139)

where σ must satisfy:

P_{0}σ = σ  (3.140)

We may translate the above two requirements as follows. The first requirement is that the restrictions S_{k,k}σ_{k} to each Γ_{k} of the single-layer potentials corresponding to the densities σ_{k} satisfy

S_{k,k}σ_{k} = f_{k} + c  (3.141)

for some constant c independent of k. The second requirement is:

∑_{k=1,..,m} ∫_{Γ_{k}} σ_{k}dθ = 0  (3.142)

There is no guarantee, however, that each σ_{k} averages to 0 on the corresponding curve Γ_{k}, which as we know is a necessary condition in the single loop case for harmonicity at infinity of the potential S_{k,k}σ_{k} if there is no boundary, and for satisfiability of the Neumann condition on the boundary if there is one. Because of this, the map S_{1}^{†}P_{0} as would be defined above by inverting Eq. 3.139 on the range of the projection P_{0}, does not quite give us the vector whose k– th component is the Dirichlet to Neumann map N_{k}(d) (either no boundary or boundary version) for the k–th component Γ_{k} of the interface.

Instead, the Dirichlet-to-Neumann map N_{k}(d) = S_{k,k}^{†}P_{k} (the pseudoinverse is now on the range of the projection P) is obtained by finding a density σ̄_{k} satisfying:

P_{k}f_{k} = P_{k}S_{k,k}σ̄_{k}.  (3.143)

This condition is equivalent to the conjunction of two statements. First, a condition on the average:

∫_{Γ_{k}} σ̄_{k}dθ = 0  (3.144)

Second, the Dirichlet condition:

S_{k,k}σ̄_{k} = f_{k} + c_{k},  (3.145)

where the constant c_{k} may depend on k.

We will now show how to obtain the desired densities σ_{k} solving Eqs. 3.139 and 3.140 from the densities σ̄_{k} solving Eqs. 3.144 and 3.145. It is clear that by solving the equation:

S_{k,k}δ_{k} = 1  (3.146)

we must then have, by Eq. 3.141, Eq. 3.145, and uniqueness for the Dirichlet problem,

σ_{k} = σ̄_{k} + (c − c_{k})δ_{k}.  (3.147)

Using Eqs. 3.142 and 3.144, we now obtain from Eq. 3.147:

c = \frac{∑_{k=1,..,m} c_{k} ∫_{Γ_{k}} δ_{k}dθ}{∑_{k=1,..,m} ∫_{Γ_{k}} δ_{k}dθ}  (3.148)

At d = 0, we can explicitly find the functions δ_{k} which satisfy Eq. 3.146. Indeed, from Eq. 3.46 we find that the δ_{k} are constants given by:

δ_{k} = \frac{2π}{log r_{0,k}}  (3.149)

Because of assumption 3.125, the δ_{k} are well-defined, as is the constant c in Eq. 3.148. This proves that P_{0}S_{1}(0) maps P_{0}(E^{h−3})^{m} onto P_{0}(E^{h−2})^{m}. Injectivity of P_{0}S_{1}(0) on P_{0}(E^{h−3})^{m} is an immediate consequence of Eq. 3.46. We conclude that P_{0}S_{1}(0) is invertible as a map from P_{0}(E^{h−3})^{m} to P_{0}(E^{h−2})^{m} and therefore that the Fredholm-Riesz-Schauder theory applies to Eq. 3.135 as described above.

We now proceed to prove injectivity of the operator on the right-hand side of Eq. 3.135. Suppose σ_{1}, ..., σ_{m} are such that for each j = 1, ..., m,

P_{0} ∑_{k=1,..,m} (−S_{j,∞} (−\frac{1}{2}I + K_{∞,∞}^{∗})^{†} K_{∞,k}^{∗} + S_{j,k}) σ_{k} = 0 (3.150)

Then for some constant c, the harmonic function u defined by the densities σ_{k} via Eq. 3.128 has constant Dirichlet datum c on each component Γ_{j} of the interface by Eq. 3.129 and 0 Neumann datum on the exterior boundary ∂Ω by Eq. 3.131. By subtracting c from u everywhere, we may assume that u vanishes on all the Γ_{j}. An application of the divergence theorem as in the single loop case now shows that u must be constant, and thus the normal derivative jumps σ_{k}/\frac{ds}{dθ} across the curves Γ_{k} are identically zero. This proves injectivity and therefore invertibility of the system in Eq. 3.135 as desired.

It follows that the Dirichlet-to-Neumann map N(d) is as given in Eq. 3.138. The analyticity of N(d) in d follows from the results in chapter 2 just as in the case of a single loop. We conclude that the Dirichlet-to-Neumann map is in the case of m loops a continuous map from (E^{h−2})^{m} to (E^{h−3})^{m} for each fixed d ∈ (E^{h})^{m}, and is analytic as a map d ↦ N(d) from (E^{h})^{m} to C ((E^{h−2})^{m}, (E^{h−3})^{m}).

3.3.2 Maximal regularity

Finally, we examine the maximal regularity issue in the multiple loop case. The strategy will be to follow the procedure used in the single loop case, namely, to show that the perturbation of the Mullins–Sekerka vector field relative to the field N(0)(κ′(0)d), where N(0) is a scalar multiple of the pseudoinverse of the single-layer integral operator with respect to functions of mean 0, is continuous from E^{4} into E^{h−3}.

More precisely, if we let N_{k}(d) denote the Dirichlet to Neumann map for the k–th component of the interface in the boundaryless case, and then combine the maps for the various loops into a single diagonal matrix operator:

N^{Δ}(d) = [ \begin{matrix}N_{1}(d) \\ & . & . & . \\ & & & & N_{m}(d) \end{matrix} ] ,  (3.151)

we will show that the linearized Mullins–Sekerka vector field differs from the field N^{Δ}(0)κ′(d) by a term which is continuous from (E^{4})^{m} into (E^{h−3})^{m}. The desired maximal regularity result will then follow from the fact that since h > 4, E^{4} is either an interpolation space strictly between E^{h−3} and E^{h}, or else is strictly below E^{h} on the E^{a} scale, together with the perturbation result from Angenent [3] which was used in the single loop case (cf. the comments preceding Theorem 3.2.2).

By Eqs. 3.44 and 3.61, the operator

(S_{1})^{†}P κ′ + ((S_{1})^{†}P )′ κ = N^{Δ}(0)κ′(d)  (3.152)

represents the linearized Mullins-Sekerka vector field corresponding to the m individual loops without any interactions among loops. The pseudoinverses here are computed relative to the projection P. Thus, the operator S_{1}^{†} maps P (E^{h−2})^{m} to P (E^{h−3})^{m}. In particular, the inversion is performed independently for the different components Γ_{k} of the interface, relative to the corresponding spaces of functions whose average on each individual Γ_{k} vanishes. The main result of section 3.2.2 implies that the linear vector field in 3.152 satisfies the MR property when d = 0.

We wish to show that the MR property also holds for the linearization of the multiloop Mullins-Sekerka vector field constructed using the Dirichlet-to-Neumann map from Eq. 3.138:

(S_{1} + B)^{†}P_{0}κ′ + ((S_{1} + B)^{†}P_{0})′ κ,  (3.153)

The dagger now denotes pseudoinverses relative to the range of the projection P_{0} rather than P, i.e. with respect to the subspaces of functions whose average over the \underline{union} of the components Γ_{k} vanishes.

Two terms must be examined in Eq. 3.153. The first, (S_{1} + B)^{†} P_{0}κ′, is dealt with as follows. We first show that this term satisfies MR if we omit B.

Recalling the definitions of the single-layer densities σ_{k} and σ̄_{k} in Eqs. 3.139, 3.140, 3.143, 3.144, we see that Eq. 3.147 may be written as follows:

S_{1}(d)^{†}P_{0} = S_{1}(d)^{†}P + W (d),  (3.154)

where W (d) is the map defined by the second term on the right-hand side of Eq. 3.147, whose range consists of constant multiples of the function δ, which is an an (E^{h})^{m} function on the interface. Clearly, W (d) therefore maps (L^{1})^{m} continuously into (E^{h})^{m}.

Thus, in particular, S_{1}^{†}P_{0} differs from the diagonal operator S_{1}^{†}P of the Dirichlet to Neumann maps for the individual curves Γ_{k} by a map which is continuous from (E^{h−2−ϵ})^{m} to (E^{h})^{m}. Therefore, the perturbation S_{1}^{†}P_{0}κ′(0) − S_{1}^{†}P κ′(0) is continuous from (E^{h−ϵ})^{m} to (E^{h})^{m}, and satisfies the property in Eq. 3.119 relative to the pair ((E^{h−3})^{m}, (E^{h})^{m}). Since S_{1}^{†}P κ′(0) has the MR property relative to ((E^{h−3})^{m}, (E^{h})^{m}), we conclude that S_{1}^{†}P_{0}κ′(0) does also.

To complete our analysis of the first term in Eq. 3.153, we must consider the effect of the operator B. The important point is that the operators which appear in B involve integration on curve i and evaluation on curve j, where i ≠ j, so that their kernels are nonsingular. Therefore, the argument used in the proof of Theorem 3.2.2 may be applied. We conclude that the map ((S_{1} + B) (0))^{†} P_{0}κ′(0)d satisfies MR.

The second term in Eq. 3.153, which we must prove the MR property for also, is

(((S_{1} + B)^{†}P_{0})′(0)d) κ(0).  (3.155)

Again, assume first that B = 0. We will show that the target map obtained by letting B = 0 in Eq. 3.155 maps E^{h−ϵ} continuously into E^{h−3} for sufficiently small ϵ > 0 and therefore satisfies the boundedness condition in Eq. 3.119 relative to the state pair (X = (E^{h−3})^{m}, Y = (E^{h})^{m}). It will follow that this operator may be added to the first term on the right-hand side of Eq. 3.153 without affecting MR.

The target map equals

((S_{1}^{†})′ (0)d) P_{0}κ(0)

We know that S_{k,k}(d) may be written in the form A_{k} + B̃_{k}(d), with A_{k} invertible as a map from E^{a−1} to E^{a} for any a > 2 and B̃_{k}(d) continuous from L^{1} to E^{h−2}, as in Eqs. 3.49 and 3.50 (we use the notation B̃_{k} rather than B_{k} here to avoid confusion with the matrix operator which appears in Eq. 3.137). It follows that the linearization ((S_{1}^{†})′(0)d) P_{0} is given by:

((A + B̃(d))^{†}P_{0})′ (0)d = −A^{†}(B̃′(0)d)A^{†}P_{0}  (3.156)

We computed the linearization of the k–th component B̃_{k}(d) of B̃(d) at d = 0 explicitly in Eq. 3.60. Recall that B̃′_{k}(0) maps d ∈ E^{h−ϵ} analytically to B̃′_{k}(0)d ∈ C(L^{1}, E^{h−ϵ}) for any sufficiently small ϵ > 0. Since each A_{k}^{†} maps E^{∞} continuously to E^{∞} and therefore to L^{1}, it follows from Eq. 3.156 that ((S_{1}^{†})′(0)d) P_{0} maps d ∈ (E^{h−ϵ})^{m} analytically to C((E^{∞})^{m}, (E^{h−1−ϵ})^{m}). Composing with the curvature κ(0), which is constant and therefore belongs to (E^{∞})^{m}, we conclude that the target map (((S_{1}^{†})′(0)d) P_{0}) κ(0) maps (E^{h−ϵ})^{m} continuously into (E^{h−1−ϵ})^{m}. Letting 0 < ϵ < 2, we see that the target map satisfies the boundedness condition in Eq. 3.119 relative to the state pair (X = (E^{h−3})^{m}, Y = (E^{h})^{m}), and the maximal regularity property therefore holds for the operator

(S_{1} + B)^{†}P_{0}κ′(0)d + ((S_{1}^{†}P_{0})′ (0)d) κ(0).  (3.157)

Finally, it may be seen that the map B defined in Eq. 3.137 may be added to the map S_{1} in the second term on the right-hand side of Eq. 3.157 without affecting MR, by showing that B is smoothing. To do this, we compute the difference between the B = 0 and the nonzero B expressions, using the fact that for small d, the operator S_{1} + B(d) is invertible, and in particular:

(S_{1} + B(d))^{†} = P_{0}(S_{1} + B(d))^{−1}P_{0}

We obtain:

(((S_{1} + B)^{†}P_{0})′ (0)d) κ(0) − ((S_{1}^{†}P_{0})′ (0)d) κ(0) =
−P_{0}S_{1}^{−1}(0)(B′(0)d)S_{1}^{−1}(0)P_{0}κ(0)  (3.158)

The operator B involves the evaluation of single-layer integrals along curves other than those supporting the single layers, in the form of the operators S_{i,j} with i ≠ j which make up the matrix operators S_{2} and S_{∞}. These operators therefore have kernels which are as smooth as the interface curve, say E^{h}, and it follows that they map L^{1} into E^{h} continuously with analytic dependence on d in E^{h}. Also appearing in B is the double-layer operator K_{∞,∞}^{∗}, whose kernel is of the form O(θ^{2})/O(θ^{2}) with both numerator and denominator in E^{h−1} and which therefore belongs to E^{h−3}. The operator K_{∞,∞}^{∗} defines a continuous map from L^{1} into E^{h−3} ⊆ L^{1} if h > 4, again with analytic dependence on d. We see that B itself therefore maps L^{1} continuously to E^{h} for each d ∈ E^{h}, and that the map d ↦ B(d) is analytic from E^{h} to C(L^{1}, E^{h}). The linearization B′(0) maps E^{h} to C(L^{1}, E^{h}) as well.

In 3.158, the operator S_{1}^{†}(0)P_{0} is continuous as a map from P_{0}(E^{a})^{m} to P_{0}(E^{a−1})^{m} for any a > 2, as may be seen by examining its components, using Eq. 3.47. Proceeding from right to left in Eq. 3.158, keeping track of the spaces involved, we see that d ∈ (E^{h})^{m} is mapped to (E^{h−1})^{m}. It now follows that B satisfies the relative boundedness property 3.119, and therefore does not affect MR relative to the pair ((E^{h−3})^{m}, (E^{h})^{m}). We conclude that the linearized Mullins-Sekerka vector field given in Eq. 3.153 satisfies MR. This completes the proof of Theorem 1.2.1 in the general case.

Bibliography

[1] L.V. Ahlfors. Complex Analysis, McGraw–Hill, 1966

[2] N. Alikakos, P. Bates, X. Chen. “Convergence of the Cahn–Hilliard equation to the Hele–Shaw model”, Archive for Rational Mechanics and Analysis, vol 128, 1994, 165–205

[3] S.B. Angenent. “Nonlinear Analytic Semiflows”, Proceedings of the Royal Society of Edinburgh, vol 115A, 1990, 91–107

[4] S.B. Angenent, M.E. Gurtin. “Anisotropic motion of a phase interface”, Journal für die reine und angewandte Mathematik, vol 446 (1994), 1–47

[5] J.B. Baillon. “Caractère borné de certains générateurs de semigroupes lineaires dans les espaces de Banach”, Comptes Rendus Acad. Sciences Paris, vol 290 (1980), 757–760

[6] P.W. Bates, X. Chen, X. Deng. “A Numerical Scheme for the Two Phase Mullins–Sekerka Problem”, Electronic Journal of Differential Equations, http://ejde.math.swt.edu/ or ftp://ejde.math.swt.edu, login:ftp, 1995

[7] J. Bricmont, A. Kupiainen, G. Lin. “Renormalization Group and Asymptotics of Solutions of Nonlinear Parabolic Equations”, Communications in Pure and Applied Mathematics, vol 47, no 6, Jun 1 1994 [8] G. Caginalp. “The dynamics of a conserved phase field system: Stefan–

like, Hele–Shaw, and Cahn–Hilliard models as asymptotic limits”, IMA Journal of Applied Mathematics, vol 44, 1990, 77–94

[9] X. Chen. “Hele–Shaw problem and area preserved curve shortening motion”, Archive for Rational Mechanics and Analysis, vol 123, no 2, 1993, 117–151

[10] X. Chen, J. Hong, F. Yi. “Existence, uniqueness, and regularity of classical solutions for the Mullins–Sekerka problem”, preprint, 1995

[11] P. Constantin, M. Pugh. “Global Solutions for Small Data to the Hele– Shaw Problem”, Nonlinearity, vol 6, 1993, 393–415

[12] G. Dore, A. Venni. “Maximal regularity for parabolic initial–boundary value problems in Sobolev spaces”, Mathematische Zeitschrift, vol 208, 1991, 297–308

[13] B. Eberhardt, G. Greiner. “Baillon’s Theorem on Maximal Regularity”, Acta Applicandae Mathematicae, vol 27, 1992, 47–54

[14] L.C. Evans, H.M. Soner, P.E. Souganidis. “Phase Transitions and Generalized Motion by Mean Curvature”, Communications on Pure and Applied Mathematics, vol 45, 1097–1123, 1992

[15] G.B. Folland. Introduction to Partial Differential Equations, Princeton, 1976

[16] M.E. Gurtin. Thermomechanics of Evolving Phase Boundaries in the Plane, Oxford Mathematical Monographs, Oxford University Press, 1993 [17] T. Ilmanen. “Convergence of the Allen–Cahn Equation to Brakke’s Motion

by Mean Curvature”, J. of Differential Geometry, vol 38 (1993), 417–461

[18] Y. Katznelson. An Introduction to Harmonic Analysis, Dover, 1976

[19] O.D. Kellogg. Foundations of Potential Theory, Dover, 1953

[20] A. Lunardi. Analytic Semigroups and Optimal Regularity in Parabolic Problems, Progress in Nonlinear Differential Equations and their Applications 16, Birkhäuser, 1995

[21] U.F. Mayer. “One–sided Mullins–Sekerka flow does not preserve convexity”, Electronic Journal of Differential Equations, http://ejde.math.swt.edu/ or ftp://ejde.math.swt.edu, login:ftp, 1993

[22] G.B. McFadden, P.W. Voorhees, R.F. Boisvert, D.I. Meiron. “A Boundary Integral Method for the Simulation of Two–Dimensional Particle Coarsening”, Journal of Scientific Computing, vol 1, no 2, 117–144, 1986

[23] P. de Mottoni, M. Schatzman. “Évolution géometrique d’interfaces”, Comptes Rendus de l’Académie des Sciences de Paris, t. 301, Série I, 453– 458, 1989

[24] P. de Mottoni, M. Schatzman. “Geometric Evolution of Developed Interfaces”, Transactions of the American Mathematical Society, vol 347, no. 7, 1533–1589, 1991

[25] W.W. Mullins, R.F. Sekerka. “Morphological stability of a particle growing by diffusion or heat flow”, Journal of Applied Physics, vol 34, 1963, 323– 329 [26] A. Pazy. Semigroups of Linear Operators and Applications to Partial Dif-

ferential Equations, Applied Mathematical Sciences 44, Springer, 1983

[27] R.L. Pego. “Front Migration in the Nonlinear Cahn–Hilliard Equation”, Proceedings of the Royal Society of London A, vol 422, 261–278, 1989

[28] O. Penrose, P.C. Fife. “Thermodynamically consistent models of phase– field type for the kinetics of phase transitions”, Physica D, vol 43, 44–62, 1990

[29] L. Simon. Partial Differential Equations, Lecture notes, 1992

[30] H. M. Soner. “Convergence of the Phase-Field Equations to the Mullins-Sekerka Problem with Kinetic Undercooling”, Archive for rational mechanics and analysis, vol 131, no 2, 139, 1995

[31] E.M. Stein (ed.) Beijing Lectures in Harmonic Analysis, Annals of Mathematics Studies 112, Princeton, 1986

[32] B.E.E. Stoth. “Convergence of the Cahn-Hilliard Equation to the Mullins-Sekerka Problem in Spherical Symmetry”, Journal of differential equations, vol 125, no 1, 154, 1996

[33] R.S. Strichartz. A guide to distribution theory and Fourier transforms, Boca Raton, CRC Press, Studies in advanced mathematics, 1994

[34] E.C. Titchmarsh. Introduction to the theory of Fourier integrals, Oxford, The Clarendon Press, 1948