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

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

2004-10-05

arxiv.org/abs/math/0410078v1

Mathematics

Analysis of PDEs

14 pages

Scientific paper

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} $.

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