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Bulletin of the London Mathematical Society Advance Access published online on September 3, 2009
Bulletin of the London Mathematical Society, doi:10.1112/blms/bdp075
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© 2009 London Mathematical Society

Hardy inequality and Lp estimates for the torsion function

M. van den Berg

Department of Mathematics
University of Bristol
University Walk
Bristol
BS8 1TW
United Kingdom

Tom Carroll

Department of Mathematics
University College Cork
Cork
Ireland
t.carroll@ucc.ie

Received 7 January 2009.

It is shown that the torsion function for an open set D in Euclidean space Rm is in L{infty}(D) if and only if the spectrum of the Dirichlet Laplacian in D is bounded away from 0. For 1 ≤ p ≤ {infty}, it is shown that the torsion function for D is in Lp(D) precisely when the distance to the boundary function is in L2p(D), if it is assumed that the Dirichlet Laplacian acting in L2(D) satisfies a strong Hardy inequality.

2000 Mathematics Subject Classification 35J25, 35P99, 58J35.

M. van den Berg was supported by The Leverhulme Trust, Research Fellowship 2008/0368.


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