Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains. (English) Zbl 1225.35077

Summary: For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain \(\Omega \) we show that solutions of the corresponding elliptic problem with Robin and thus in particular with Neumann boundary conditions are Hölder continuous up to the boundary for sufficiently \(L^{p}\)-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell-Robin boundary conditions is well-posed on \(C \bar {\varOmega}\).


35J25 Boundary value problems for second-order elliptic equations
35B65 Smoothness and regularity of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
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