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# 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: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) (KAR id:31253)

 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. Official URLhttp://dx.doi.org/10.1007/s00209-006-0055-6

## 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 10.1007/s00209-006-0055-6 Q Science > QA Mathematics (inc Computing science) > QA299 Analysis, CalculusQ Science > QA Mathematics (inc Computing science) > QA377 Partial differential equations Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science Ian Wood 04 Oct 2012 10:32 UTC 16 Nov 2021 10:09 UTC https://kar.kent.ac.uk/id/eprint/31253 (The current URI for this page, for reference purposes) https://orcid.org/0000-0001-7181-7075
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