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 u Omega = inf u is an element of D-1,D-2(Omega) Vu not equivalent to 0 integral(Omega)vertical bar del u vertical bar(2) dx/integral(Omega) V vertical bar u vertical bar(2) dx, where Omega is a domain in R-N,R- and V is a nonzero nonnegative function that may have singularities on partial derivative 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/vertical bar X vertical bar (2).