An isoperimetric inequality for the first Steklov-Dirichlet Laplacian eigenvalue of convex sets with a spherical hole

We prove the existence of a maximum for the first Steklov–Dirichlet eigenvalue in the class of convex sets with a fixed spherical hole, under volume constraint. More precisely, if Ω = Ω 0 ∖ ¯¯¯¯ B R 1 , where B R 1 is the ball centered at the origin with radius R 1 > 0 and Ω 0 ⊂ R n , n ≥ 2 , is an open, bounded and convex set such that B R 1 ⋐ Ω 0 , then the first Steklov–Dirichlet eigenvalue σ 1 ( Ω ) has a maximum when R 1 and the measure of Ω are fixed. Moreover, if Ω 0 is contained in a suitable ball, we prove that the spherical shell is the maximum.