Regularity of solutions of linear second order elliptic and parabolic boundary value problems on Lipschitz domains

For a linear, strictly elliptic second order differential operator in divergence form with bounded, measurable coefficients on a Lipschitz domain Ω 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 Lp-regular right-hand sides. From this we deduce that the parabolic problem with Robin or Wentzell–Robin boundary conditions is well-posed on .