Topics in Potential Theory: Quadrature Domains, Balayage and Harmonic Measure. TOMAS SJÖDIN Doctoral Thesis Stockholm, Sweden 2005 TRITA-MAT-05-MA-05 ISSN 1401-2278 KTH Matematik ISRN KTH/MAT/DA--05/03--SE SE-100 44 Stockholm ISBN 91-7178-009-2 SWEDEN Akademisk avhandling som med tillstånd av Kungl. Tekniska Högskolan framlägges till oentlig granskning för avläggande av teknologie doktorsexamen tisdagen den 31 maj 2005, klockan 10.00 i D3, Kungl. Tekniska Högskolan, Lindstedtsvägen 5, Stockholm. °c Tomas Sjödin, maj 2005 Tryck: Intellecta DocuSys AB iii Abstract In this thesis, which consists of ve papers (A,B,C,D,E), we are interested in questions related to quadrature domains. Among the problems studied are the possibility of changing the type of measure in a quadrature identity (from complex to real and from real signed to positive), properties of partial balayage, which in a sense can be used to generate quadrature domains, and mother bodies which are closely related to inversion of partial balayage. These three questions are discussed in papers A,D respectively B. The rst of these questions (when trying to go from real signed to positive measures) leads to the study of approximation in the cone of positive harmonic functions. These questions are closely related to properties of the harmonic measure on the Martin boundary, and this relationship leads to the study of harmonic measures on ideal boundaries in paper E. Some other approaches to the same problem also lead to some extent to the study of properties of classical balayage in paper C. iv Sammanfattning I denna avhandling, som består av fem artiklar (A,B,C,D,E), är vi in- tresserade av frågor som hänger samman med kvadraturområden. Bland de problem som studeras är möjligheten att byta typ av mått i en kvadraturiden- titet (från komplext till reellt och från reellt med tecken till positivt), egen- skaper hos partiell balayage, som i viss mening kan användas för att generera kvadraturområden, och moderkroppar, som är närbesläktade med inversion av partiell balayage. Dessa frågor studeras i artikel A,D respektive B. Den första av dessa frågor (när man försöker gå från reellt med tecken till positivt mått) leder till approximation i konen av positiva harmoniska funktioner. Dessa frågor är besläktade med egenskaper hos det harmoniska måttet på den så kallade Martin-randen, och denna relation ledde till studiet av harmoniska mått på allmänna ideala ränder i artikel E. Några andra försök att angripa detta problem ledde i viss mening till studiet av klassisk balayage i artikel C. ACKNOWLEDGMENTS: v Acknowledgments: Since I have spent almost ve years working on this thesis there are so many I wish to thank that it will simply not be possible to mention them all individually. First of all my family, and then especially my darling Åsa, but also a lot of my colleagues here at KTH. In particular my fellow graduate students which gives the department its wonderfull atmosphere. But the one person to whom I owe the most is of-course my supervisor and mentor Björn Gustafsson. I met Björn about seven years ago when I took a topology course he gave, and later I decided to do my master thesis together with him. When choosing a supervisor it is only after a little while that you can be sure you made a good choice, and after nishing my master thesis it was clear to me that Björn was excellent, not only because of his broad knowledge (it was seldom he was left without an answer even to questions far from his own areas of research), but also for his point of view towards mathematics and research in general. So it was natural to continue working with him on my doctoral thesis. In particular Björn has always been very generous with his time, listening to my ideas and read my manuscripts carefully, giving valuable suggestions (mathematical as well as typographical). I dedicate this thesis to Björn, and hope he has gained something from all the hours he spent on it. In any case it has meant very much to me. Tomas Sjödin ACKNOWLEDGMENTS: vii Contents Contents v 1. Introduction 1 Paper A Quadrature Identities and Deformation of Quadrature Domains. (This paper was rst published as [12].) 1. Introduction 239 2. Basic denitions and notation 240 3. Deformation of quadrature domains 243 4. Complex to real identities 247 5. Real to positive identities 251 References 255 Paper B Mother Bodies of Certain Domains in the Complex Plane. 1. Introduction 1 2. Basic notation and denitions 2 3. Some continuity results 4 3.1. Continuity, method 1 6 3.2. Continuity, method 2 9 4. Existence and uniqueness of a mother body for the ellipse 11 5. Another existence result for the ellipse 15 References 16 Paper C Balayage 1. Introduction 1 2. Notation 2 3. Problem and Results 3 4. Another question concerning balayage 8 References 10 Paper D Partial Balayage Classical and Probabilistic Approaches. 1. Introduction 1 2. Partial Balayage 2 3. Probabilistic part 7 References 9 viii Paper E Relations between Harmonic Measures on Elements in the Compactication Lattice of a Euclidean Domain. 1. Introduction 2 1.1. History of the article 2 1.2. Contents of the paper 4 2. Preliminary material and basic notation 5 2.1. The Compactication Lattice 6 2.2. Potential-theoretic preliminaries 10 3. Some Important Boundaries 20 3.1. The Martin Boundary 20 3.2. The Wiener boundary 21 3.3. The Universal Wiener boundary 26 3.4. On the poles of Minimal Harmonic Functions 28 4. Connections between harmonic measures, part 1 31 5. Connections between harmonic measures, part 2 37 5.1. Vector-lattices 38 5.2. Comparison of two lattice structures 40 6. Poissonian domains and related topics 45 7. Harmonic measures and approximation in HhP (Ω) 51 8. Structure of ∂MEΩ (∂M Ω) 54 References 56 Appendix Potential Theory 1. About the Document 2 2. Preliminaries 2 2.1. Notation 2 2.2. Calculus 4 2.3. Topology 4 2.4. The Compactication Lattice 7 3. Harmonic, sub- and superharmonic functions 11 4. Green functions and Potentials 21 5. Polar sets, Reduced functions and Capacity 28 5.1. Polar sets 28 5.2. Reduced functions 30 5.3. Capacity 32 6. Energy 40 7. PWBh-method and the Dirichlet problem 42 8. Fine topology 53 9. The Martin boundary 61 References 82 Chapter 1 Introduction and summary In this introduction to the thesis we will summarize some of the material studied in the ve articles it consists of, and motivate the questions studied in them. I will try to avoid technicalities as far as possible. All ve articles have originated in one way or another from questions related to quadrature domains. Before we introduce this concept we need to say a few things about classical potential theory, which is the main tool used to study them in this thesis. I wish to point out that all potential theory needed may be found in the appendix of the thesis. It however does not contain anything non-standard, and the same material may be found in several books. My personal favorite reference, where most of the material may be found, is the book Classical Potential Theory by Armitage and Gardiner, which is more detailed than my appendix. In particular the last two chapters in the appendix are based almost entirely on that book. Classical potential theory is concerned with the Laplacian dierential operator ∆ dened in Euclidean n−space Rn by ∂2 ∂2 ∆ := 2 + ... + 2 . ∂x1 ∂xn If Ω ⊂ Rn is open, then a function h ∈ C2(Ω) such that ∆h = 0 in Ω, is called harmonic in Ω. The vector space of all harmonic functions in Ω is denoted H(Ω). One of the fundamental results about these are that a function h ∈ C(Ω) is harmonic in Ω if and only if Z Z 1 1 h(a) = hdm = hdσ ∀B(a, r) ⊂⊂ Ω, m(B(a, r)) B(a,r) σ(∂B(a, r)) ∂B(a,r) (1.1) 1 2 CHAPTER 1. INTRODUCTION AND SUMMARY where B(a, r) is the open ball with center a and radius r, ⊂⊂ denotes that it is com- pactly contained in Ω, and the measures m, σ are volume (Lebesgue) respectively hypersurface measure. It follows from (1.1) that h ∈ C2(Ω). Let us briey indicate why (1.1) is equivalent to harmonicity. To do this we recall the divergence theorem Z Z ∇ · F dm = F · Ndσ,ˆ Ω ∂Ω where F is any C1 vector-eld in a neighourhood of Ω and ∂Ω is a C1 bound- ary. The vector Nˆ denotes the outward pointing unit normal to ∂Ω, and ∇ = 2 (∂/∂x1, . , ∂/∂xn). If h ∈ C (Ω) we may look at the function Z 1 Fh(a, r) := hdσ, σ(∂B(a, r)) ∂B(a,r) for a, r such that B(a, r) ⊂⊂ Ω. Now ∂ Fh(a, r + ε) − Fh(a, r) Fh(a, r) = lim ∂r ε&0 ε à Z Z ! 1 1 = lim hdσ − hdσ ε&0 σ(∂B(a, r + ε)) ∂B(a,r+ε) σ(∂B(a, r)) ∂B(a,r) Z Z 1 1 = ∇h · Ndσˆ = ∆hdm, σ(∂B(a, r)) ∂B(a,r) σ(∂B(a, r)) B(a,r) since ∇ · ∇ = ∆. From this two things follow immediately. First of all, if h is harmonic, then the last integral is zero, so Fh(a, r) is independent of r, and by continuity, when we let r tend to zero we see that we have h(a) = Fh(a, r). Integrating this with respect to r we get that (1.1) holds if h is harmonic.