  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
$C(\bar{\Omega})$.
