article

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

Authors: Yehuda Pinchover and Kyril Tintarev

Issue: Volume 54 (2005), Issue 4, 1061-1074

Abstract:

$The paper studies the existence of minimizers for Rayleigh quotients % $\mu_{\Omega}=\inf\displaystyle\int_{\Omega}|\nabla % u|^2\dx/\displaystyle\int_{\Omega}V{|u|^2}\dx$, \[ \mu_{\Omega}=\inf\frac{\displaystyle\int_{\Omega}|\nabla u|^2\dx}{\displaystyle\int_{\Omega}V{|u|^2}}\dx, \] 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 $V(x)=1/|x|^2$.$