Maximal Lp-regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Omega, both with the following two domains of definition: D-1(Delta) = {u is an element of W-1,W-p(Omega) : Delta u is an element of L-p(Omega), Bu = 0}, or D-2(Delta) = {u is an element of W-2,W-p(Omega) : Bu = 0}, , where B is the boundary operator. We prove that, under certain restrictions on the range of p, these operators generate positive analytic contraction semigroups on L-p(Omega) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Omega is bounded and convex and 1 < p <= 2, the Laplacian with domain D (2)(Delta) has the maximal regularity property, as in the case of smooth domains. In the last part, we construct an example that proves that, in general, the Dirichlet-Laplacian with domain D (1)(Delta) is not even a closed operator.