Elliptic and parabolic problems with Robin boundary conditions on Lipschitz domains

Abstract

It is shown that for a wide class of quasi-linear elliptic equations in divergence form, including the p-Laplace equation, on Lipschitz domains the solution is Hölder continuous up to the boundary provided the right hand side is sufficiently regular. From this it is deduced that the associated parabolic problem is well-posed in the space of continuous functions. More precisely, the part of the operator in the space of continuous functions generates a strongly continuous non-linear contraction semigroup.