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

Wood, Ian (2007) Maximal L^p -regularity for the Laplacian on Lipschitz domains. Mathematische Zeitschrift, 255 (4). pp. 855-875. ISSN 0025-5874. (doi:https://doi.org/10.1007/s00209-006-0055-6) (The full text of this publication is not currently available from this repository. You may be able to access a copy if URLs are provided)

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
Subjects:	Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, Calculus Q Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations
Divisions:	Faculties > Sciences > School of Mathematics Statistics and Actuarial Science
Depositing User:	Ian Wood
Date Deposited:	04 Oct 2012 10:32 UTC
Last Modified:	28 Jul 2014 08:05 UTC
Resource URI:	https://kar.kent.ac.uk/id/eprint/31253 (The current URI for this page, for reference purposes)