Elliptic and Parabolic Robin Problems on Lipschitz Domains: Hölder Continuity of Solutions of Elliptic Problems and Generation of Nonlinear Semigroups on the Space of Continuous Functions

The author studies regularity of quasi-linear elliptic and parabolic equations with Robin boundary conditions on Lipschitz domains. He shows that solutions of elliptic problems with Neumann boundary conditions are Hölder continuous up to the boundary under very mild assumptions which resemble the optimal assumptions for interior Hölder regularity. Using the result for Neumann problems, the author obtains global Hölder regularity also for Robin problems. Finally, the elliptic results are used to show that the corresponding operator on the space of continuous functions is m-accretive and thus generates a strongly continuous nonlinear semigroup.