We consider operators of the form \mathcal L=-L-V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain \Omega\subset R^n with Dirichlet boundary conditions. We allow the boundary of \Omega to be made of various pieces of different codimension. We assume that \mathcal L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator \mathcal 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.
IMPROVING L^2 ESTIMATES TO HARNACK INEQUALITIES / Filippas, S; Moschini, Luisa; Tertikas, A.. - In: PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY. - ISSN 0024-6115. - 99 (part 2):(2009), pp. 326-352. [10.1112/plms/pdp002]
IMPROVING L^2 ESTIMATES TO HARNACK INEQUALITIES.
MOSCHINI, Luisa;
2009
Abstract
We consider operators of the form \mathcal L=-L-V, where L is an elliptic operator and V is a singular potential, defined on a smooth bounded domain \Omega\subset R^n with Dirichlet boundary conditions. We allow the boundary of \Omega to be made of various pieces of different codimension. We assume that \mathcal L has a generalized first eigenfunction of which we know two-sided estimates. Under these assumptions we prove optimal Sobolev inequalities for the operator \mathcal 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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
