Maximal L p -regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary conditions on bounded Lipschitz domains Ω, both with the following two domains of definition: D1(Δ)={u∈W1,p(Ω):Δu∈Lp(Ω), Bu=0}D1(Δ)={u∈W1,p(Ω):Δu∈Lp(Ω), Bu=0} , or D2(Δ)={u∈W2,p(Ω):Bu=0}D2(Δ)={u∈W2,p(Ω):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 (Ω) which implies maximal regularity for the corresponding Cauchy problems. In particular, if Ω is bounded and convex and 1<p≤21<p≤2 , the Laplacian with domain D 2(Δ) 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(Δ) is not even a closed operator.