## Quasilinear elliptic and parabolic Robin problems on Lipschitz domains.(English)Zbl 1268.35021

Summary: We prove Hölder 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 $$\mathrm{C}(\overline \Omega)$$ provided that the coefficients satisfy a mild monotonicity condition. More precisely, we show that the realization of the elliptic operator in $$\mathrm{C}(\overline\Omega)$$ is $$m$$-accretive and densely defined. Thus it generates a non-linear strongly continuous contraction semigroup on $$\mathrm{C}(\overline\Omega)$$.

### MSC:

 35B65 Smoothness and regularity of solutions to PDEs 35R05 PDEs with low regular coefficients and/or low regular data 35J25 Boundary value problems for second-order elliptic equations 35K15 Initial value problems for second-order parabolic equations
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