Layer Potential Operators for Domains with Minimal Boundary Regularity

The main purpose of this Master's dissertation is to define and to study the single layer potential and the double layer potential operators for domains in Euclidean space with minimal boundary regularity, namely domains of class $C^1$ and Lipschitz domains. Such operators are used in mathematical analysis to solve, for example, the homogeneous Dirichlet and Neumann problems for Laplace's equation with $L^p$-boundary data in a bounded domain in $\mathbb{R}^n$: this is achieved by transforming the original boundary value problem into a certain integral equation on the boundary of the domain. Under the assumption that the domain is - at least - of class $C^1$, we are able to solve this integral equation establishing the invertibility of the boundary integral operator via the classical Fredholm theory. If, on the other hand, the domain is Lipschitz, the boundary integral operator may not be compact anymore; thus the invertibility of the solution operator is much harder to prove. The first chapter of this thesis focuses on domains in $\mathbb{R}^n$: at the beginning we develop the theory of Lipschitz domains and then (domains of class $C^k$, $k \in \mathbb{N}^* \cup \{\infty\}$, being a particular case of Lipschitz domains, see theorem \ref{teo:t1.2.1}), we give the corresponding definitions and statements for the class $C^k$, $k \in \mathbb{N}^* \cup \{\infty\}$. The results presented here are at the same time preliminary to the second chapter and interesting in their own right. The second chapter is devoted to layer potential operators and their boundary counterparts. Precisely, we introduce the single layer potential, the double layer potential and the boundary layer potential operators.