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

This paper deals with the eigenvalue problem for the operator L=−Δ−x⋅∇L=−Δ−x⋅∇ with Dirichlet boundary conditions. We are interested in proving the existence of a set minimizing any eigenvalue λkλk of L   under a suitable measure constraint suggested by the structure of the operator. More precisely we prove that for any c>0c>0 and k∈Nk∈N the following minimization problemmin⁡{λk(Ω):Ωquasi-openset,∫Ωe|x|2/2dx≤c} has a solution.