In this paper we study the first Steklov-Laplacian eigenvalue with an internal fixed spherichal obstacle. We prove that the spherical shell locally maximizes the first eigenvalue among nearly spherical sets when both the internal ball and the volume are fixed.

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the L^{\infty} and L^p norms, for p\ge 1 , of the torsion function, seen as a functional on a bounded simply connected open set \Omega\subset\mathbb{R}^n , and prove that the balls are critical shapes for these functionals, when the volume of \Omega is preserved.

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In this paper we prove the existence of a maximum for the first Steklov-Dirichlet Laplacian eigenvalue in the class of convex sets with a fixed spherical hole under volume constraint. More precisely, if Ω " Ω 0 zB R1 , where B R1 is the ball centered at the origin with radius R 1 ą 0 and Ω 0 Ă R n , n ě 2, is an open bounded and convex set such that B R1 Ť Ω 0 , then the first Steklov-Dirichlet Laplacian eigenvalue σ 1 pΩq 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.

In this paper we consider PDE's problems involving the anisotropic Laplacian operator, with Robin boundary conditions. By means of Talenti techniques, widely used in the last decades, we prove a comparison result between the solutions of the above-mentioned problems and the solutions of the symmetrized ones. As a consequence of these results, a Bossel-Daners type inequality can be shown in dimension 2. MSC 2010: 35B51 -35G20 -35G30 -35J60 -35P15