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

In this paper 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 $\Omega=\Omega_0 \setminus \bar{B}{R_1}$,where$B{R_1}$istheballcenteredattheoriginwithradius$R_1>0$and$\Omega_0\subset$\mathbb{R}$^n$,$n\geq 2$, is an open bounded and convex set such that $B_{R_1}⋐{\mathrm{\Omega }}_0$, then the first Steklov-Dirichlet eigenvalue ${\sigma }_1(\mathrm{\Omega })$ has a maximum when $R_1$ and the measure of $\mathrm{\Omega }$ are fixed. Moreover, if ${\mathrm{\Omega }}_0$ is contained in a suitable ball, we prove that the spherical shell is the maximum.