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Proceedings of the London Mathematical Society Advance Access published online on March 3, 2009
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdp002
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© 2009 London Mathematical Society

Improving L2 estimates to Harnack inequalities

Stathis Filippas

Department of Applied Mathematics
University of Crete
71409 Heraklion
Greece
Institute of Applied and Computational Mathematics
FORTH
71110 Heraklion
Greece
filippas@tem.uoc.gr

Luisa Moschini

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate
University of Rome "La Sapienza"
00161 Rome
Italy
moschini@dmmm.uniroma1.it

Achilles Tertikas

Department of Mathematics
University of Crete
71409 Heraklion
Greece
Institute of Applied and Computational Mathematics
FORTH 71110 Heraklion
Greece

Received 16 January 2008. Revision received 30 October 2008.

We consider operators of the form L = – L – V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain {Omega} sub Rn with Dirichlet boundary conditions. We allow the boundary of {Omega} to be made of various pieces of different codimension. We assume that L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator L, we show that it generates an intrinsic ultracontractive semigroup and finally we derive a parabolic Harnack inequality up to the boundary as well as sharp heat kernel estimates.

2000 Mathematics Subject Classification 35K65, 26D10 (35K20, 35B05).


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