Existence of minimizers for eigenvalues of the Dirichlet-Laplacian with a drift

This paper deals with the eigenvalue problem for the operator $L = −Delta − x cdot abla$ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue $lambda_k$ of $L$ under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any $c > 0$ and $kin mathbb{N}$ the following minimization problem $$ minleft{lambda_k(Omega): Omega extrm{quasi-open set}, int_Omega e^{|x|^2/2}dxle c ight} $$ has a solution.