Improving $L^2$ estimates to Harnack inequalities

Details Improving $L^2$ estimates to Harnack inequalities Improving $L^2$ estimates to Harnack inequalities

2009-11-04

arxiv.org/abs/0911.0947v1

Mathematics

Analysis of PDEs

Scientific paper

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.

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