Quasilinear elliptic and parabolic Robin problems on Lipschitz domains |
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Authors: | Robin Nittka |
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Institution: | 1. Max Planck Institute for Mathematics in the Sciences, Inselstr. 22, 04103, Leipzig, Germany 2. Institute of Applied Analysis, University of Ulm, 89069, Ulm, Germany
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Abstract: | 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 ${\mathrm{C}(\overline{\Omega})}$ . |
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