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

Wood, I. (2007) Maximal L^p -regularity for the Laplacian on Lipschitz domains. Mathematische Zeitschrift, 255 (4). pp. 855-875. ISSN 0025-5874. (The full text of this publication is not available from this repository)

## Abstract

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.

Item Type: Article Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations Faculties > Science Technology and Medical Studies > School of Mathematics Statistics and Actuarial Science Ian Wood 04 Oct 2012 10:32 23 Jan 2013 09:00 http://kar.kent.ac.uk/id/eprint/31253 (The current URI for this page, for reference purposes)