Boundary value problems with measures for elliptic equations with singular potentials

Details Boundary value problems with measures for elliptic equations with singular potentials Boundary value problems with measures for elliptic equations with singular potentials

2010-07-15

arxiv.org/abs/1007.2482v1

Mathematics

Analysis of PDEs

Contient un Appendice d'A. Ancona intitul\'e A necessary condition for the fine regularity of a boundary point with respect to

Scientific paper

We study the boundary value problem with Radon measures for nonnegative solutions of $L_Vu:=-\Delta u+Vu=0$ in a bounded smooth domain $\Gw$, when $V$ is a locally bounded nonnegative function. Introducing some specific capacity, we give sufficient conditions on a Radon measure $\gm$ on $\prt\Gw$ so that the problem can be solved. We study the reduced measure associated to this equation as well as the boundary trace of positive solutions. In the appendix A. Ancona solves a question raised by M. Marcus and L. V\'eron concerning the vanishing set of the Poisson kernel of $L_V$ for an important class of potentials $V$.

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