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Quasilinear elliptic and parabolic Robin problems on Lipschitz domains

  • Robin NittkaEmail author
Article

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 \({\mathrm{C}(\overline{\Omega})}\) .

Mathematics Subject Classification (1991)

Primary 35B65 35R05 Secondary 35J25 35K15 

Keywords

Second order quasi-linear elliptic equations Lipschitz domains Robin boundary conditions Hölder regularity Unbounded coefficients Parabolic equations Non-linear semigroup Space of continuous functions Wentzell–Robin boundary conditions 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Max Planck Institute for Mathematics in the SciencesLeipzigGermany
  2. 2.Institute of Applied AnalysisUniversity of UlmUlmGermany

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