The Dirichlet Problem for Second-Order Divergence Form Elliptic Operators with Variable Coefficients: The Simple Layer Potential Ansatz

Abstract

We investigate the Dirichlet problem related to linear elliptic second-order partial differential operators with smooth coefficients in divergence form in bounded connected domains of ${\mathbb{R}}^m$ ($m\ge \mathrm{3}$) with Lyapunov boundary. In particular, we show how to represent the solution in terms of a simple layer potential. We use an indirect boundary integral method hinging on the theory of reducible operators and the theory of differential forms.