A Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition

We prove a Faber-Krahn inequality for the Laplacian with drift under Robin boundary condition, provided that the β parameter in the Robin condition is large enough. The proof relies on a compactness argument, on the convergence of Robin eigenvalues to Dirichlet eigenvalues when β goes to infinity, and on a strict Faber-Krahn inequality under Dirichlet boundary condition. We also show the existence and uniqueness of drifts v satisfying some L^∞ constraints and minimizing or maximizing the principal eigenvalue of -Δ+v·∇ in a fixed domain and with a fixed parameter β>0 in the Robin condition.