# Simple Layer Potentials on Lipschitz Surfaces: An Asymptotic Approach

This work is devoted to the equation 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.

Contents

Introduction
The Simple Layer Potential
Boundary Integral Methods
Main Results
Paper 1: Riesz Potential Equations in Local Lp-spaces
1 Introduction
2 Preliminary Np-estimates
3 Proof of Theorem 1.5
4 Proof of Theorem 1.3
5 Proof of Theorem 1.7
6 Proof of Theorem 1.1
7 Proof of Theorem 1.8
8 Applications to Weighted Function Spaces
Paper 2: A Fixed Point Theorem in Locally Convex Spaces
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
3 Applications
3.1 A First Order Differential Equation
3.2 A Second Order Differential Equation
3.3 A Pseudodiﬀerential Equation
Paper 3: An Asymptotic Approach to Simple Layer Potentials on Lipschitz Surfaces
1 Introduction
2 The Simple Layer Potential
2.1 Riesz Potentials on RN
2.2 Singular Integral Operators
2.3 Diﬀerentiation of S
2.4 Approximation of S by Riesz Potentials
2.5 Seminorm Estimates
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
4 Proof of the Main Results
4.1 Existence of Solutions
4.2 Uniqueness of Solutions
5 Results under Various Assumptions on ω
5.1 The Case When ω = ω0 is a Constant
5.2 The Case When Nω + ω0≥ 0
Paper 4: Invariant Properties of Riesz Potentials
1 Introduction
2 Preliminary Deﬁnitions and Notation
3 Invariant Properties of Riesz Potentials
3.1 Riesz Potentials of Radial Functions
3.2 Riesz Potentials of Spherical Functions
4 Invariant Properties of the Operator R
4.1 Estimates for Spherical Functions

Author: Thim, Johan