Link¨oping Studies in Science and Technology Dissertations, No 1235
Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach Johan Thim
Division of Applied Mathematics Department of Mathematics Link¨oping 2009
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Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach c 2009 Johan Thim, unless otherwise noted. Copyright Matematiska institutionen Link¨opings universitet SE-581 83 Link¨ oping, Sweden
[email protected] Link¨oping Studies in Science and Technology Dissertations, No 1235 ISBN 978-91-7393-709-2 ISSN 0345-7524 http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-16280
Printed by LiU-Tryck, Link¨ oping 2009
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Abstract This work is devoted to the equation ˆ u(y) dS(y) = f (x), N −1 S |x − y|
x ∈ S,
(1)
where S is the graph of a Lipschitz function ϕ on RN with small Lipschitz constant, and dS is the Euclidian surface measure. The integral in the left-hand side is referred to as a simple layer potential and f is a given function. The main objective is to find a solution u to this equation along with estimates for solutions near points on S. Our analysis is carried out in local Lp -spaces and local Sobolev spaces, and the estimates are given in terms of seminorms. In Paper 1, we consider the case when S is a hyperplane. This gives rise to the classical Riesz potential operator of order one, and we prove uniqueness of solutions in the largest class of functions for which the potential in (1) is defined as an absolutely convergent integral. We also prove an existence result and derive an asymptotic formula for solutions near a point on the surface. Our analysis allows us to obtain optimal results concerning the class of right-hand sides for which a solution to (1) exists. We also apply our results to weighted Lp - and Sobolev spaces, showing that for certain weights, the operator in question is an isomorphism between these spaces. In Paper 2, we present a fixed point theorem for a locally convex space X , where the topology is given by a family {p( · ; α)}α∈Ω of seminorms. We study the existence and uniqueness of fixed points for a mapping K : DK → DK defined on a set DK ⊂ X . It is assumed that there exists a linear and positive operator K, acting on functions defined on the index set Ω, such that for every u, v ∈ DK , p(K (u) − K (v) ; α) ≤ K(p(u − v ; · ))(α),
α ∈ Ω.
Under some additional assumptions, one of which is the existence of a fixed point for the operator K + p(K (0) ; · ), we prove that there exists a fixed point of K . For a class of elements satisfying K n (p(u ; · ))(α) → 0 as n → ∞, we show that fixed points are unique. This class includes, in particular, the solution we construct in the paper. We give several applications, proving existence and uniqueness of solutions for two types of first and second order nonlinear differential equations in Banach spaces. We also consider pseudodifferential equations with nonlinear terms. In Paper 3, we treat equation (1) in the case when S is a general Lipschitz surface and 1 < p < ∞. Our results are presented in terms of Λ(r), which is the Lipschitz constant of ϕ on the ball centered at the origin with radius 2r. Estimates of solutions to (1) are provided, which can be used to obtain knowledge about behaviour near a point on S in terms of seminorms. We also show that solutions to (1) are unique if they are subject to certain growth conditions. Examples are given when specific assumptions are placed on Λ. The main tool used for both existence and uniqueness is the fixed point theorem from Paper 2. In Paper 4, we collect some properties and estimates of Riesz potential operators, and also for the operator that was used in Paper 1 and Paper 3 to invert the Riesz potential of order one on RN , for the case when the density function is either radial or has mean value zero on spheres. It turns out that these properties define invariant subspaces of the respective domains of the operators in question.
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Acknowledgements First and foremost, I’d like to thank both my advisers Vladimir Kozlov and Bengt Ove Turesson for their invaluable guidance during the last five years; this would have been impossible without their help. Moreover, going back to my undergraduate studies at Lule˚ a University of Technology, many thanks to Lech Maligranda and Reinhold N¨ aslund who both taught me much and had a huge impact on my development. Going further back, I extend my gratitude to Tore Hansen for providing a very stimulating environment in so many ways: granting access to “his” school premises at all times and giving inspiration and preparation for further studies. Thanks also to Hans-Eric Hellberg for being very supportive in the use of computer labs at nonstandard hours. Those years were very important in cementing my interest in technical subjects and mathematics. And last, but certainly not least, a huge thanks to my parents for their undying support of all my escapades throughout the years. Whether it consisted of being supportive when I was listening to excessively loud music, blowing stuff up, hacking away in assembler all night, or machine-gunning down monsters in DooM, they’ve been impressively understanding. Thank you all! Link¨ oping, 12 January 2009. Johan Thim
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Popul¨ arvetenskaplig introduktion Bakgrund Antag att vi st¨ alls inf¨ or en situation d¨ ar vi beh¨over hitta en harmonisk funktion i ett omr˚ ade, d¨ ar det ¨ ar f¨ oreskrivet vad funktionen skall ha f¨or v¨arden p˚ a randen. Detta kallas f¨ or Dirichlets problem f¨or Laplaces ekvation. Exempel p˚ a n¨ar problem av den h¨ ar typen kan uppst˚ a finns i fysiken, d¨ar till exempel gravitationspotentialen eller den elektriska potentialen i fria rymden b˚ ada ¨ar harmoniska funktioner. I v¨ armeledning dyker det h¨ar problemet ocks˚ a upp om man vill veta hur temperaturf¨ ordelningen ser ut i ett objekt n¨ar temperaturen p˚ a randen ¨ar given. Kanske mer intressant, ur ett matematiskt perspektiv, ¨ar att Laplaces ekvation ¨ ar det enklaste exemplet p˚ a en elliptisk ekvation. Full f¨orst˚ aelse f¨or detta prototypproblem ¨ ar allts˚ a viktig f¨or att f¨orst˚ a hur mer komplicerade ekvationer kan hanteras. I specifika fall g˚ ar det ibland att explicit skriva upp hur en l¨osning ser ut. Om omr˚ adet till exempel ¨ ar ett klot, kan man direkt beskriva l¨osningen med hj¨alp av poissonintegralen. N¨ ar omr˚ adet ¨ar mer komplicerat, ¨ar det inte l¨angre uppenbart hur man kan visa att l¨ osningar existerar, men olika metoder finns f¨or att behandla situationen. Ett s¨ att ¨ ar att formulera om problemet som en integralekvation p˚ a randen av omr˚ adet, n˚ agot som kan ˚ astadkommas till exempel med hj¨ alp av s˚ a kallade lagerpotentialer. Om omr˚ adet ¨ar glatt (inga h¨orn eller andra elakheter), har de operatorer, som dessa integraler ger upphov till, trevliga egenskaper i form av kontinuitet och kompakthet p˚ a l¨ampliga rum. Problem uppst˚ ar dock om man l¨ attar p˚ a kravet att randen skall vara sn¨all, d˚ a dessa egenskaper inte l¨ angre ¨ ar uppenbara, eller ¨overhuvudtaget g¨aller. En av metoderna f¨ or att formulera problemet som en integralekvation brukar kallas f¨or den direkta metoden. Ide´en ¨ ar att givet ett dirichletvillkor p˚ a randen ˚ aterskapa motsvarande neumannvillkor, vilket ¨ar v¨ardet av normalderivatan p˚ a randen, och sedan anv¨ anda detta f¨ or att representera l¨osningen med hj¨alp av k¨anda integralidentiteter. S˚ alunda st¨ alls vi inf¨or problemet att l¨osa en integralekvation, vilket mynnar ut i att invertera en enkellagerpotential. Detta objekt beskrivs av en integral ¨ over randen p˚ a omr˚ adet, d¨ar integranden inneh˚ aller en (svag) singularitet som f¨ orsv˚ arar arbetet. Givet en funktion, som best¨ams av dirichletvillkoret, s¨ oker vi nu en t¨athet f¨or potentialen s˚ a att potentialen i fr˚ aga sammanfaller med den givna funktionen. Denna t¨athet ¨ar allts˚ a den s¨okta normalderivatan. Vi ¨ ar intresserade av att genomf¨ora detta under s˚ a svaga f¨oruts¨ attningar som m¨ ojligt p˚ a ing˚ aende funktioner, och att f˚ a fram uppskattningar f¨ or t¨ atheten n¨ ara en punkt p˚ a randen. Sammanfattning Den h¨ar avhandling kretsar kring problemet att invertera en enkellagerpotential given p˚ a en yta som inte a ¨r glatt utan bara har lipschitzregularitet. Detta till˚ ater vissa typer av h¨ orn och andra kantigheter. Analysen utf¨ors i rummet som best˚ ar av funktioner som a ¨r lokalt p-summerbara utanf¨or origo, d¨ar strukturen
vi best¨ams av en familj seminormer indexerade med en positiv parameter. Seminormen av en funktion kan tolkas som “funktionsv¨ardet” p˚ a ett visst avst˚ and fr˚ an origo. Huvudsyftet ¨ ar att ge skarpa uppskattningar f¨or l¨osningarna till potentialekvationen i termer av seminormerna. Uppskattningarna i fr˚ aga beskrivs av integraloperatorer som verkar p˚ a funktioner definierade p˚ a halvaxeln (det vill s¨aga indexm¨ angden f¨ or seminormerna). Rent konkret kommer ytan vi behandlar att vara grafen f¨or en lipschitzkontinuerlig funktion i flera variabler, d¨ ar den enda inskr¨ankningen vi g¨or ¨ar att anta att lipschitzkonstanten ¨ ar tillr¨ ackligt liten. F¨orst betraktar vi fallet d˚ a ytan ¨ar ett hyperplan, och i detta fall kan potentialen ovan reduceras till en klassisk rieszpotential av ordning ett. Vi visar att den operator, som rieszpotentialen ger upphov till, ¨ ar injektiv p˚ a den st¨ orsta m¨ ojliga definitionsm¨angden (om vi kr¨aver absolutkonvergens av integralen i fr˚ aga). Vi introducerar sedan, med hj¨alp av riesztransformer, en operator som ¨ ar inversen till rieszpotentialoperatorn f¨or sn¨alla funktioner, och visar att den g˚ ar att utvidga till en h¨ogerinvers f¨or en st¨orre klass av funktioner. Skarpa uppskattningar f¨or h¨ogerinversen ger de efters¨okta uppskattningarna f¨ or l¨ osningar till rieszpotentialekvationen. Vidare betraktar vi n˚ agra invarianta underrum till dessa operatorer. Viss f¨orsiktighet ¨ ar n¨ odv¨ andig vid hanterandet av b˚ ade inversen till rieszpotentialoperatorn och derivatan av potentialen, d˚ a de integraler som dyker upp bara ¨ar villkorligt konvergenta (s˚ a kallade singul¨ara integraler). Med hj¨alp av v¨alk¨anda tekniker ¨ ar det dock m¨ ojligt att utnyttja kancellationsegenskaper hos k¨arnan i operatorn genom att tolka integralen som ett gr¨ansv¨arde d¨ar vi integrerar symmetriskt kring singulariteten. Denna process blir ¨annu k¨ansligare p˚ a en lipschitzyta, men kan hanteras tillfredsst¨ allande tack vare resultat av Calder´on, Coifman med flera f¨ or cauchyk¨ arnan. F¨or att visa motsvarande satser n¨ ar ytan inte l¨angre ¨ar ett hyperplan formulerar vi om fr˚ agest¨ allningen till en fixpunktsekvation, och h¨arleder sedan en fixpunktssats f¨ or rum d¨ ar strukturen ges av seminormer. Satsen ¨ar mer generell ¨an n¨odv¨ andigt f¨ or att l¨ osa potentialekvationen, men visar p˚ a en intressant struktur n¨ar en “besv¨ arlig” operator kan ¨ overskattas av en positiv och linj¨ar operator som verkar p˚ a funktioner definierade p˚ a indexm¨angden f¨or den familj seminormer som anv¨ ands. Vi utnyttjar detta f¨or att l¨osa tv˚ a stycken olinj¨ara differentialekvationer av ordning ett och tv˚ a i ett Banachrum och f¨or att studera pseudodifferentialekvationer med olinj¨ ara termer.
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Contents Introduction The Simple Layer Potential . . . . . . . . . . . . . . . . . . . . . . Boundary Integral Methods . . . . . . . . . . . . . . . . . . . . . . Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Paper 1: Riesz Potential Equations in Local Lp -spaces
1 1 2 6
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1 Introduction
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2 Preliminary Np -estimates
19
3 Proof of Theorem 1.5
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4 Proof of Theorem 1.3
24
5 Proof of Theorem 1.7
25
6 Proof of Theorem 1.1
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7 Proof of Theorem 1.8
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8 Applications to Weighted Function Spaces
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Paper 2: A Fixed Point Theorem in Locally Convex Spaces
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1 Introduction 2 Main Results 2.1 The Operator K . . . . . . . . . . . . . . . . . . 2.2 The Operator K . . . . . . . . . . . . . . . . . . 2.3 Existence of Fixed Points . . . . . . . . . . . . . 2.4 Uniqueness of Fixed Points . . . . . . . . . . . . 2.5 Error Estimates . . . . . . . . . . . . . . . . . . . 2.6 Comparison with Banach’s Fixed Point Theorem
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3 Applications 56 3.1 A First Order Differential Equation . . . . . . . . . . . . . . . 56 3.2 A Second Order Differential Equation . . . . . . . . . . . . . 58 3.3 A Pseudodifferential Equation . . . . . . . . . . . . . . . . . . 62
viii Paper 3: An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces
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1 Introduction 2 The 2.1 2.2 2.3 2.4 2.5
Simple Layer Potential Riesz Potentials on RN . . . . . . . . . . Singular Integral Operators . . . . . . . Differentiation of S . . . . . . . . . . . . Approximation of S by Riesz Potentials Seminorm Estimates . . . . . . . . . . .
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3 Reduction to a Fixed Point Problem 3.1 The Fixed Point Equation . . . . . . . . . . . . . . 3.2 A Fixed Point Theorem in Locally Convex Spaces 3.3 Verification of (K1) and (K2) . . . . . . . . . . . . 3.4 An Auxiliary Equation . . . . . . . . . . . . . . . . 3.5 Properties of gω . . . . . . . . . . . . . . . . . . . . 3.6 Existence of a Solution to the Auxiliary Equation . 3.7 Verification of (K3), (K4), (K 1), and (K 2) . . . 3.8 Existence of a Fixed Point . . . . . . . . . . . . . . 3.9 Uniqueness of Fixed Points . . . . . . . . . . . . .
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4 Proof of the Main Results 96 4.1 Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Uniqueness of Solutions . . . . . . . . . . . . . . . . . . . . . 102 5 Results under Various Assumptions on ω 103 5.1 The Case When ω = ω0 is a Constant . . . . . . . . . . . . . 103 5.2 The Case When N ω + ω 0 ≥ 0 . . . . . . . . . . . . . . . . . . 104
Paper 4: Invariant Properties of Riesz Potentials
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1 Introduction
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2 Preliminary Definitions and Notation
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3 Invariant Properties of Riesz Potentials 112 3.1 Riesz Potentials of Radial Functions . . . . . . . . . . . . . . 113 3.2 Riesz Potentials of Spherical Functions . . . . . . . . . . . . . 120 4 Invariant Properties of the Operator R 122 4.1 Estimates for Spherical Functions . . . . . . . . . . . . . . . . 123
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Introduction “How many integrals does this contain?” –B.O. The answer is 755 (only counting explicit symbols).
The Simple Layer Potential Throughout the next hundred pages or so, we will study the equation ˆ U (Q) dS(Q) = F (P ), P ∈ S, N −1 S |P − Q|
(1.1)
where S is a Lipschitz surface in RN +1 and dS is the Euclidian surface measure. The object in the left-hand side is the simple layer potential of the Laplacian in RN +1 . For a given right-hand side F in this equation, we will find a solution U along with estimates, in terms of seminorms, that can be used to gain information about the behaviour of the solution near a point on the surface. This is the main goal of the thesis. The analysis is carried out for U in local Lp -spaces and F in local Sobolev spaces. The surface S is the graph in RN +1 of a Lipschitz function ϕ on RN with small Lipschitz constant. We assume that ϕ(0) = 0. The function ϕ is characterised by a one-variable function Λ such that for r > 0, Λ(r) is the optimal constant in |ϕ(x) − ϕ(y)| ≤ Λ(r)|x − y| for all x, y ∈ RN such that |x|, |y| ≤ 2r. The function Λ : [0, ∞) → [0, ∞) is increasing and bounded: sup Λ(r) = Λ0 < ∞, (1.2) r≥0
where Λ0 is the (global) Lipschitz constant of ϕ. For x ∈ RN , let Φ(x) = (x, ϕ(x)) be a parametrization of S. We also define u(x) = U (Φ(x)) and f (x) = F (Φ(x)), and let S be the operator induced by the left-hand side in (1.1): p ˆ u(y) 1 + |∇ϕ(y)|2 Su(x) = dy, x ∈ RN . (1.3) N −1 RN |Φ(x) − Φ(y)| Thus, the equation we are concerned with, can be formulated as Su(x) = f (x), x ∈ RN .
(1.4)
One motivation for studying equation (1.4) is that this integral equation appears in the direct approach to the Dirichlet problem for the Laplace operator in a domain in RN +1 . We start by giving some background to boundary integral methods.
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Boundary Integral Methods Let Ω be a bounded, simply connected domain in RN +1 with smooth boundary. We always assume that N ≥ 2. A function v, harmonic in Ω and smooth up to the boundary, can be represented as ˆ ∂v(Q) ∂E(P, Q) v(P ) = − E(P, Q) dS(Q) (1.5) v(Q) ∂nQ ∂nQ ∂Ω for P ∈ Ω, where E(P, Q) = c|P − Q|1−N is the fundamental solution to the Laplacian, and nQ is the outwards normal on ∂Ω. We define the simple layer potential Su by ˆ Su(P ) = E(P, Q)u(Q) dS(Q), P ∈ RN +1 \ ∂Ω, ∂Ω
and the double layer potential Du ˆ ∂E(P, Q) u(Q) dS(Q), P ∈ RN +1 \ ∂Ω. Du(P ) = ∂nQ ∂Ω If ∂Ω is regular, say C 1,α , then both Su and Du are defined for P on the boundary ∂Ω if u ∈ C(∂Ω); these values are the direct values of the potentials. Furthermore, the natural extension of Su is continuous on RN +1 . One can show that Du(Q) has a limit (Du)+ (P ) as Q ∈ Ω approaches the boundary, and that this limit satisfies (Du)+ (P ) =
1 u(P ) + Du(P ), P ∈ ∂Ω. 2
(1.6)
Similarly, one can show that ∂Su (P ) = ∂nP
ˆ u(Q) ∂Ω
∂E(P, Q) dS(Q) ∂nP
is defined for both P ∈ Ω and P ∈ ∂Ω, and that Su possesses a regular normal derivative (∂Su/∂n)+ on ∂Ω that satisfies ∂Su ∂Su 1 (P ) = − u(P ) + (P ), P ∈ ∂Ω. (1.7) ∂nP + 2 ∂nP The regular normal derivative of a function v ∈ C 1 (Ω) at a point P on the boundary ∂Ω is defined by the limit, if it exists, of ∂v/∂nP (P − tnP ) as t → 0+ . For proofs of the above, see, e.g., Maz’ya [25]. The Direct Approach Suppose that we are given complete Cauchy data, i.e., that v(P ) = φ(P ) (the Dirichlet data) and that ∂v(P )/∂nP = ψ(P ) (the Neumann data), for P ∈ ∂Ω,
3 where φ, ψ ∈ C(∂Ω). We tacitly assume that the integral of ψ over the boundary is equal to zero. To satisfy these boundary conditions, we must have ˆ ∂E(R, Q) φ(P ) = lim φ(Q) − E(R, Q)ψ(Q) dS(Q), P ∈ ∂Ω. Ω3R→P ∂Ω ∂nQ Taking into account the continuity of φ and of the simple layer potential, along with the continuity of ψ and the “jump” relation in (1.6), we obtain the equation ˆ ∂E(P, Q) 1 φ(P ) = φ(Q) − E(P, Q)ψ(Q) dS(Q), P ∈ ∂Ω. (1.8) 2 ∂nQ ∂Ω If only the the Dirichlet data φ is known, we can solve (1.8) for the corresponding Neumann data ψ, and use this to find a harmonic function that satisfies the boundary condition by means of equation (1.5). Defining f by ˆ 1 ∂E(P, Q) dS(Q) − φ(P ), P ∈ ∂Ω, f (P ) = φ(Q) ∂nQ 2 ∂Ω and letting u(Q) = cψ(Q), we arrive at equation (1.1): ˆ u(Q) dS(Q) = f (P ), P ∈ ∂Ω. N −1 ∂Ω |P − Q|
(1.9)
If, for example, ∂Ω is C 2,α , then S −1 exists and maps L2 (∂Ω) continuously into H −1 (∂Ω); see, e.g., Maz’ya [25, p. 147]. In Costabel [5], the operator S is considered as an operator between certain fractional Sobolev spaces on ∂Ω, where ∂Ω is merely assumed to be Lipschitz. Variational techniques are used to study solvability. If, instead, the Neumann date is known, one solves (1.8) for the corresponding Dirichlet data. This gives rise to an equation of the form 1 φ(P ) − Dφ(P ) = f (P ), P ∈ ∂Ω. 2
(1.10)
For example, if ∂Ω is C 1,α , this equation can be solved by applying the Fredholm alternative, since in this case, D is a compact operator from C(∂Ω) into C(∂Ω). Equation (1.10) is of the same type that will show up when using the indirect approach, so we now turn to consider this. The Indirect Approach Another approach to solving the boundary value problem is to exploit the fact that a solution in the form of a layer potential must satisfy (1.6) or (1.7) for the Dirichlet or Neumann problem, respectively. Given Dirichlet data on ∂Ω, we then solve for a density u in the equation 1 u(P ) + Du(P ) = φ(P ), P ∈ ∂Ω, 2
(1.11)
4 and use this solution to obtain a harmonic function in terms of the double layer potential Du which satisfies this boundary condition. Similarly, given Neumann data ψ on ∂Ω (expressed as the regular normal derivative), we can solve for a density u in 1 u(P ) − D∗ u(P ) = −ψ(P ), P ∈ ∂Ω, (1.12) 2 where D∗ is the operator with kernel (∂/∂nP )E(P, Q), and obtain a harmonic function Su which satisfies this boundary condition. Equation (1.12) corresponds to the “jump” of the simple layer potential (1.7). Observe also that D∗ is the (formal) adjoint operator of D. As in the direct approach, one can employ Fredholm’s alternative to solve (1.11) and (1.12) since both D and D∗ are compact operators on C(∂Ω) if ∂Ω is, e.g., C 1,α . This is a consequence of that the respective kernels can be estimated by a constant times |P − Q|−(N −1−α) . These operators are also compact on Lp (∂Ω) for 1 < p < ∞, something that is exploited in, for example, Folland [12], together with the fact that Lp -solutions are continuous on ∂Ω if the boundary data belongs to C(∂Ω). For another detailed solution of these equations, see Mikhlin [28]. The Exterior Problem In the previous discussion, we only considered the interior Dirichlet and Neumann problems, that is, finding a function that is harmonic in Ω and that satisfies the respective boundary conditions. One can also consider the corresponding exterior problems, which consists of finding a function that is harmonic in Ωc . Boundary data is still given on ∂Ω, and a large part of the methodology is the same. However, some assumptions on the behaviour of solutions for points near infinity are necessary. See, e.g., Folland [12] or Mikhlin [28]. Boundary Data in Lp (∂Ω) We now consider the interior Dirichlet and Neumann problems when the boundary data is only assumed to belong to Lp (∂Ω) in more detail. Obviously, some care is necessary when defining in what sense a function solves these problems. The non-tangential maximal function Mβ u, for β > 1, of a function u defined on Ω, is given on ∂Ω by Mβ u(P ) = sup{ |u(Q)| : Q ∈ Ω, |P − Q| < β dist(Q, ∂Ω) }, P ∈ ∂Ω. Solving Dirichlet’s problem amounts to finding a harmonic function in Ω, that converges non-tangentially (i.e., limits are taken in cones with positive aperture) to the given boundary data φ at almost every point of ∂Ω, and such that Mβ u belongs to Lp (∂Ω) whenever β > 1. Similarly, Neumann’s problem is solved if we, for given boundary data ψ, can find a harmonic function u such that Mβ (∇u) belongs to Lp (∂Ω) and ∇u(Q) · nP → ψ(P ) as Q → P nontangentially. The vector nP is the outwards normal at P ∈ ∂Ω (which exists almost everywhere on ∂Ω if ∂Ω is Lipschitz). See, e.g., Dahlberg [8] for details.
5 Historical Notes on Boundary Value Problems for Laplace’s Equation in Lipschitz- and C 1 -Domains In an article from 1979, Dahlberg (see [7] and also a technical report from 1977– 78) showed that the Dirichlet problem has a solution if the boundary is either C 1 and the boundary data belongs to Lp (∂Ω) for 1 < p < ∞, or the boundary is Lipschitz and the boundary data belongs to Lp (∂Ω), where 2 − ε ≤ p < ∞ for some ε > 0 that depends on the domain. The method used was based on analysis of the harmonic measure, and could not be applied to the Neumann problem. The limit on the range of p is sharp: if 1 < p < 2, there exists a Lipschitz domain for which the Dirichlet problem has no solution (in the sense above), something that Dahlberg pointed out in [7], referencing his earlier work with harmonic measures [6]; see also Section 2(b) in Kenig [18]. In 1978, Fabes, Jodeit, and Rivi`ere [11] (see also the work together with A.P. Calder´on and C.P. Calder´ on [2]) solved both the Dirichlet and Neumann problem when ∂Ω is a C 1 surface and the given boundary data belongs to Lp (∂Ω) for 1 < p < ∞. This was accomplished by means of layer potentials, utilising an earlier result by Calder´ on [1], which showed that the Cauchy integral on a Lipschitz curve is bounded on Lp for 1 < p < ∞ if the Lipschitz constant is small. They proved that the layer potentials are compact on Lp (∂Ω), and could use Fredholm theory to solve both problems (similarly to what was done earlier for C 1,α domains). However, if ∂Ω is merely Lipschitz, then the layer potentials do not necessarily define compact operators (see Fabes, Jodeit, and Lewis [10], or Kenig [18] for counterexamples). In 1979, Jerison and Kenig derived a simplified proof of Dahlberg’s result (see [14], [15]). Moreover, in [16], they also solved the Neumann problem for p = 2. However, the general Lp -theory was left open. Verchota, in his PhD thesis from 1982, showed that the layer potential operators are still invertible on L2 (∂Ω) in the Lipschitz case (see Verchota [34]). He used a generalisation of Calder´ on’s result about the Cauchy integral by Coifman, McIntosh, and Meyer [3]. This result removed the restriction on the size of the Lipschitz constant. He could then recover the known solvability results for L2 (∂Ω) by means of layer potentials. In 1984, Dahlberg and Kenig solved the Neumann problem for boundary data in Lp (∂Ω) with 1 < p < 2 + ε, which similarly with the Dirichlet case is the optimal range of p; see [9]. They also showed that both the solutions to the Neumann problem and the solutions to the Dirichlet problem could be obtained by the method of layer potentials. Other References and Methods An overview of boundary integral equations can be found in, for example, Maz’ya [25]. Jerison and Kenig [17] also gives a presentation of the theory of boundary value problems in Lipschitz domains. In Taylor [33], one can find a fairly short proof of the Lp -bound of the Cauchy integral on Lipschitz curves. The proof is based on the work of Coif-
6 man, Jones, and Semmes [4], and employs techniques from complex analysis (the Kobe-Bieberbach distortion theorem for instance). Taylor then develops the layer potential theory for the Dirichlet problem in Lipschitz domains, also allowing equations with variable coefficients. For a nice presentation of singular operators of the type necessary for handling the layer potentials, see Meyer and Coifman [27], where a more general class of Calder´on–Zygmund operators is treated. This book also includes some results concerning the layer potentials on Lipschitz domains. A summary of the results about the Dirichlet and Neumann problems in nonsmooth domains, and also of its history, can be found in, e.g., Kenig [19]. We also refer to Maz’ya and Shaposhnikova [26] and references therein. As was mentioned earlier, it is also possible to solve these problems by other methods. For instance, in Costabel [5], variational techniques are employed to show that there exists solutions to elliptic problems when the layer potentials are considered as operators on certain fractional Sobolev spaces on ∂Ω, and the boundary is merely Lipschitz. Similarly, variational techniques are used in Hsiao and Wendland [13] to solve elliptic problems by means of boundary integral equations.
Main Results Throughout the thesis, we work with local Lp -spaces and local Sobolev spaces on RN , where we tacitly assume that N ≥ 2. Estimates are given in terms of seminorms: Np (u ; r) =
1 rN
ˆ p
|u(x)| dx
1/p , r > 0, 1 ≤ p < ∞.
r≤|x|
