Existence of minimizers for Schrödinger operators under domain perturbations

Yehuda Pinchover Technion Institute of Technology, Israel

In this talk we shall discuss the existence of minimizers for Rayleigh quotients μ_{Ω} = inf \frac{∫_{Ω} |∇u|^{2}}{∫_{Ω} V |u|^{2}} under domain perturbations, here Ω is a domain in ℝ^{N}, and V is a nonzero nonnegative function that may have singularities on ∂Ω. 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}} .

This is a joint work with Kyril Tintarev.