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 $μ_Ω=\inf\frac{\int_Ω|\nabla u|^2}{\int_ΩV{|u|^2}} $, where $Ω$ is a domain in $\mathbb{R}^N$, and $V$ is a nonzero nonnegative function that may have singularities on $\partialΩ$. As a model for our results one can take $Ω$ to be a Lipschitz cone and $V$ to be the Hardy potential $V(x)=\frac{1}{|x|^2} $.