Existence of minimizers for Schrodinger operators under domain perturbations with application to Hardy's inequality

Abstract

The paper studies the existence of minimizers for Rayleigh quotients ${\mu }_{\Omega }=\inf \frac{{\int }_{\Omega }|\nabla u{|}^2}{{\int }_{\Omega }V|u{|}^2}$, where $\Omega$ is a domain in ${\mathbb{R}}^N$, and $V$ is a nonzero nonnegative function that may have singularities on $\partial \Omega$. As a model for our results one can take $\Omega$ to be a Lipschitz cone and $V$ to be the Hardy potential $V(x)=\frac{1}{|x{|}^2}$.

14 pages