Solvability of the Dirichlet, Neumann and the regularity problems for parabolic equations with Hölder continuous coefficients

Abstract

We establish the $L^2$-solvability of Dirichlet, Neumann and regularity problems for divergence-form heat (or diffusion) equations with Hölder-continuous diffusion coefficients, on bounded Lipschitz domains in ${\mathbb{R}}^n$. This is achieved through the demonstration of invertibility of the relevant layer-potentials which is in turn based on Fredholm theory and a new systematic approach which yields suitable parabolic Rellich-type estimates.