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POTENTIAL THEORY and HARMONIC FUNCTIONS a Thesis Submitted to Kent State University in Partial Fulfillment of the Requirements F

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POTENTIAL THEORY and HARMONIC FUNCTIONS a Thesis Submitted to Kent State University in Partial Fulfillment of the Requirements F POTENTIAL THEORY AND HARMONIC FUNCTIONS A thesis submitted to Kent State University in partial fulfillment of the requirements for the degree of Master of Science by Hebah S. Alhwaitiy December, 2015 Thesis written by Hebah S. Alhwaitiy B.S.,Tabouk University, 2009 M.S., KENT STATE UNIVERSITY, 2015 Approved by Dr. Benjamin Jaye , Advisor Dr. Andrew Tonge , Chair, Department of Mathematical Sciences Dr. James L. Blank , Dean, College of Arts and Sciences ii TABLE OF CONTENTS LIST OF FIGURES . iv LIST OF TABLES . v Acknowledgements ............................... vi 1 Introduction .................................. 1 2 Potential Theory and Harmonic Functions . 3 2.1 Harmonic Functions .......................... 3 2.2 Subharmonic Function ........................ 7 2.3 Proof of Theorem (2.1.5) ....................... 14 2.4 Polar sets and Hausdorff measure . 22 2.5 Harmonic Measure and Green;Functions . 32 2.6 Hausdorff measure and harmonic measure . 39 BIBLIOGRAPHY . 46 iii Acknowledgements I would like to thank Kent State University and Department of Mathematical Seiences for letting me fulfill my dream of being a student here. I will forever be thankful to my committe members- Dr. Dmitry Ryabogin , and Dr. Artem Zvavitch for their encouragement, helpful career advice, and suggestions in general. I would like to express my sincere gratitude to my advisor Dr. Benjamin Jaye for the continuous support my thesis study and related research for his patience, motivation, and immense knowledge. His guidance helped me in all the time of research and writting of this thesis. I could not have imagined having a better advisor and mentor for my master study. I wish to express my sincere thanks to my parents, sisters, and brothers. They always supporting me and encourging me with their best wishes. I must express my very profound gratitude to my husband, Abdulrahman for all of his love and support without which I could not have managed to complete my thesis. I would like to express my gratitude and appreciation to my friends for their support, guidance, love and generosity. I am especially thankful to my the best friend Rawan. She has continually encouraged me to perform as strongly as I can. The product of this research paper would not be possible without all of them. iv CHAPTER 1 Introduction The purpose of this thesis is to see how measure theoretic ideas aid the study of harmonic functions in very general domains. We shall describe a general solution of the Dirichlet problem, and see how this gives rise to Harmonic Measure. Many of the results in this expository thesis are taken from [1]. The main goal in the first three sections is to introduce a wide class of a domain in which we can solve the Dirichlet problem for harmonic function. First, we consider the special case when the domain is a disc, where one can find a solution directly using the Poisson integral. To solve the Dirichlet problem in a general domain, we appeal to the Perron method, which makes use of subharmonic functions. A certain notion of exceptional set arises in solving the Dirichlet problem: the polar sets. In section 4 we introduce some Hausdorff measure and describe how the polar sets have Hausdorff dimension zero. In the fifth section, we use the solution of the Dirichlet problem to produce a special measure called Harmonic Measure. We study the properties of harmonic measure using apparatus from measure theory. The harmonic measure of a domain is closely connected to the Green function, and we will prove an integral formula for the Green function in terms of the harmonic measure. We use the same technique of [1]. 1 In the last section, we introduce two results that relate the harmonic meaure to Hausdorff measure. We follow the technique of [2] with a modification that is resulted of changing the type of a domain. We shall only consider the special case of the Cantor set. 2 CHAPTER 2 Potential Theory and Harmonic Functions 2.1 Harmonic Functions Definition 2.1.1. Let U ⊂ C be an open set. A function h : U ! R is said to be harmonic if h 2 C2(U) and ∆h = 0 on U, where the Laplace operator ∆ is given by @2h @2h ∆h := + : @x2 @y2 We say that we can solve the Dirichlet problem in a domain D ⊂ C, which is a connected open set, if for every φ 2 C(@D), we can find a unique function u : D ! R satisfying (i) u is harmonic in D, and (ii) limz2D: z!ζ u(z) = φ(ζ) for each ζ 2 @D. We want to solve the Dirichlet problem in a wide class of domains. Let0s first consider the special case when D is a disc B(w; ρ) = fz 2 C : jz − wj < ρg. We now introduce the notions of the Poisson Kernel P and Poisson integral PBφ. Definition 2.1.2. (i) The Poisson kernel P : B(0; 1) × @B(0; 1) ! R is defined by ζ+z 1−|zj2 P (z; ζ) := Re( ζ−z ) = jζ−zj2 (jzj < 1; jζj = 1): 3 (ii) If B = B(w; ρ) and φ : @B ! R is a Lebesgue-integrable function, then its Poisson integral PBφ : B ! R is defined by 1 R 2π z−w iθ iθ PBφ(z) := 2π 0 P ( ρ ; e )φ(w + ρe )dθ (z 2 B) We will now discuss the Dirichlet problem for the disc. Theorem 2.1.3. (Dirichlet Problem for the disc) Let B = B(w; ρ). Then for φ 2 C(@B), the function PBφ is harmonic in B, and satisfies limz!ζ PBφ(z) = φ(ζ) for ζ 2 @B; z 2 B. Thus, PBφ solves the Dirichlet problem on B. Proof. Suppose w = 0 and ρ = 1, so that B = B(0; 1). Then Z 2π 1 iθ iθ PBφ(z) = P (z; e )φ(e )dθ 2π 0 Z 2π iθ 1 e + z iθ = Re( iθ )φ(e )dθ 2π 0 e + z Z 2π iθ 1 e + z iθ = Re iθ φ(e )dθ 2π 0 e − z then it is the real part of a holomorphic function of z. Hence it is harmonic on B. Indeed, let h = Ref where f is holomorhic on B. Set f = h + i` then by cauchy- Riemann equations we have that @h @` @h @` @x = @y and @y = − @x Therefore, @2h @2h @2` @2` + = − = 0 @x2 @y2 @x@y @y@x 4 Thus h is harmonic function. We now need to prove that limz!ζ PBφ(z) = φ(ζ) Z 2π 1 iθ iθ jPBφ(z) − φ(ζ)j = P (z; e )(φ(e ) − φ(ζ))dθ 2π 0 1 Z 2π ≤ P (z; eiθ)jφ(eiθ) − φ(ζ)jdθ; 2π 0 where we have used that P (z; ξ) ≥ 0 if z 2 B and that 1 Z 2π P (z; eiθ)dθ = 1: 2π 0 Fix ζ and let > 0. Since φ is a continuous function on the whole of @B, then for every ζ0 2 @B there exist δ > 0 such that jφ(ζ0) − φ(ζ)j < whenever jζ0 − ζj < δ: Hence, 1 Z 1 Z 2π P (z; eiθ)jφ(eiθ) − φ(ζ)jdθ ≤ P (z; eiθ)dθ 2π jeiθ−ζj<δ 2π 0 Z 2π eiθ + z = Re( iθ )dθ 2π 0 e − z Z 2π eiθ + z = Re( iθ dθ) 2π 0 e − z Z ζ + z 1 = Re( dζ) 2π jζj=1 ζ − z ζi 1 Z 2 1 = Re( ( − )dζ) 2πi jζj=1 ζ − z ζ = Re(2 − 1) = . Next we shall show that there exists δ0 > 0 such that 0 jz − ζj < δ ) sup P (z; ζ0) < . ζ02@B: jζ0−ζ|≥δ 5 Indeed, 2 2 ζ0 + z ζ0 − z jζ0j − jzj + (ζ0z − ζ0z) ( ) × ( ) = 2 ζ0 − z ζ0 − z jζ0 − zj 2 2 jζ0j − jzj + 2i=(ζ0z) = 2 jζ0 − zj So by Definition (2.1.2), 2 2 2 ζ0 + z jζ0j − jzj 1 − jzj P (z; ζ0) = Re( ) = 2 = 2 : ζ0 − z jζ0 − zj jζ0 − zj Where jzj < 1; jζ0j = 1 δ δ Now we have ζ fixed, and jζ0 − ζj > δ, if jz − ζj < 2 then jz − ζ0j > 2 . Therfore, 1 − jzj2 4 P (z; ζ0) = 2 ≤ 2 (1 − jzj)(jzj + 1): jζ0 − zj δ 0 8δ0 0 But since jzj + 1 ≤ 2 and jzj > 1 − jz − ζj > 1 − δ , we have P (z; ζ0) ≤ δ2 < if δ is chosen small enough in terms of δ. Consequently, if jz − ζj < δ0, then 1 Z 1 Z 2π P (z; eiθ)jφ(eiθ) − φ(ζ)jdθ ≤ jφ(eiθ) − φ(ζ)jdθ 2π jeiθ−ζ|≥δ 2π 0 1 Z 2π ≤ jφ(eiθ)jdθ + jφ(ζ)j : 2π 0 Combining these facts, we deduce that if jz − ζj < δ0 then 1 Z 2π 1 Z 2π P (z; eiθ)jφ(eiθ) − φ(ζ)jdθ ≤ (1 + jφ(eiθ)jdθ + jφ(ζ)j): 2π 0 2π 0 Consequently we deduce that 1 R 2π iθ jPBφ(z) − φ(ζ)j ≤ (1 + 2π 0 jφ(e )jdθ + jφ(ζ)j), from which we conclude that PBφ(z) ! φ(ζ) as z ! ζ. 6 The class of domains that we shall consider the general Dirichlet problem need to have \thick" boundaries. To quantify this we shall introduce the notions of Polar set. Definition 2.1.4. A subset E ⊂ C is called polar if RR log jz − wjdµ(z)dµ(w) = −∞ for every finite non-zero compactly supported Borel measure µ with supp(µ) ⊂ E. We say that a property holds nearly everywhere (n.e.) if it holds everywhere outside some polar set. We shall show later (in Section 2.4) that any set of finite s-dimensional Hausdorff measure, s 2 (0; 2], is non-polar.
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