Singular Integrals and Elliptic Boundary Problems on Regular Semmes-Kenig-Toro Domains

Abstract

We develop the theory of layer potentials and related singular integral operators as a tool to study a variety of elliptic boundary problems on a family of domains introduced by Semmes [105, 106] and Kenig and Toro[64–66], which we call regular Semmes–Kenig–Toro domains. This extends the classic work of Fabes, Jodeit, and Rivière in several ways. For one, the class of domains that are locally graphs of functions whose gradients have vanishing mean oscillation (VMO1 domains), which in turn contains the class of C1 domains. In addition, we study not only the Dirichlet and Neumann boundary problems but also a variety of others. Furthermore, we treat not only constant coefficient operators but also operators with variable coefficients, including operators on manifolds.