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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Nittka, R. Quasilinear elliptic and parabolic Robin problems on Lipschitz domains. Nonlinear Differ. Equ. Appl. 20, 1125–1155 (2013). https://doi.org/10.1007/s00030-012-0201-2
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DOI: https://doi.org/10.1007/s00030-012-0201-2