Maximal L p -regularity for the Laplacian on Lipschitz domains

I.Wood: Maximal Lp-regularity for the Laplacian on Lipschitz domains, Math. Z., 255, 4 (2007), 855-875.

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} , or D2(?)={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 L p (?) which implies maximal regularity for the corresponding Cauchy problems. In particular, if ? is bounded and convex and 1<p?2 , the Laplacian with domain D 2(?) 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(?) is not even a closed operator.

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