Anya Hanson: Solvability of Dirichlet and Neumann Boundary Value Problems on C^{1,α} Domains

Abstract

This thesis investigates the solvability of the Dirichlet and Neumann boundary value problems for bounded \(C^{1,α}\) domains. Through introducing layer potentials and proving “jump relations” at the boundary, resulting solvability criteria is formulated in terms of operators and thus are investigated using the theory of compact operators and the Fredholm Alternative. The results of the extension of this problem to Lipschitz domains is then introduced and compared to the \(C^{1,α}\) case.