Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

Abstract

We prove H\"older continuity up to the boundary for solutions of quasi-linear degenerate elliptic problems in divergence form, not necessarily of variational type, on Lipschitz domains with Neumann and Robin boundary conditions. This includes the $p$-Laplace operator for all $p \in (1,\infty)$, but also operators with unbounded coefficients. Based on the elliptic result we show that the corresponding parabolic problem is well-posed in the space $C(\bar{\Omega})$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in $C(\bar{\Omega})$ is m-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on $C(\bar{\Omega})$.