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

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} $.