Maximal L^p -regularity for the Laplacian on Lipschitz domains

We consider the Laplacian with Dirichlet or Neumann boundary

conditions on bounded Lipschitz domains ?, both with the following two domains of

definition:D1(?) = {u ? W1,p(?) : ?u ? Lp(?), Bu = 0}, orD2(?) = {u ? W2,p(?) :

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 Lp(?) which implies maximal regularity for the corresponding Cauchy problems.

In particular, if ? is bounded and convex and 1 < p ? 2, the Laplacian with domain

D2(?) 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 D1(?) is not even a closed operator.