DOMAIN VARIATIONS OF THE FIRST EIGENVALUE VIA A STRICT FABER-KRAHN TYPE INEQUALITY

For d ≥ 2 and (Formula Presented), we prove a strict Faber-Krahn type inequality under polarizations for the first eigenvalue λ1(Ω) of the p-Laplace operator on a bounded Lipschitz domain (Formula Presented) with mixed boundary conditions. We apply this inequality to the obstacle problems on domains of the form Ω\O, where(Formula Presented) is an obstacle. Under some geometric assumptions on Ω and O, we prove the strict monotonicity of λ1(Ω \ O) with respect to certain translations and rotations of O in Ω.