Sannipoli, Rossano (2022) Sharp estimates for eigenvalues and torsional rigidity of linear and nonlinear operators. [Tesi di dottorato]

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This doctoral Thesis deals with linear and nonlinear elliptic problems with different types of boundary conditions, with particular focus on the Steklov, Dirichlet and Robin ones. The main aim is to establish spectral isoperimetric inequalities in suitable classes of sets and quantitative estimates in terms of torsion, perimeter and measure. After a wide introduction, where it is described the state of art and an overview of the main problems studied, the Thesis is divided into four Chapters. Chapter 1 contains basic definitions and preliminary results that are fundamental in the whole dissertation. Chapter 2 deals with Shape Optimization problem of the first Steklov-Dirichlet or Steklov-Robin eigenvalue for the Laplacian in annular domains. In the Steklov-Dirichlet case, the analysis focuses firstly on the class of nearly spherical sets and afterwards in the convex case. Existence results and characterization of the optimal shapes are given. Moreover, new estimates for the first eigenvalue are proved in terms of geometrical quantities. In the Steklov-Robin case, the main properties of the first eigenvalue and its asymptotic behavior are studied. Chapter 3 deals with some linear and nonlinear torsion problem with Robin boundary conditions. In the nonlinear case a Talenti comparison result is proved for the solutions to the anisotropic Laplacian. Moreover, a Bossel-Daners type inequality is proved in dimension two. In the linear case it is shown that the balls are critical shapes for the $L^p$ and L^{\infty}$ norms of the torsion function. Chapter 4 focuses on sharp and quantitative estimates for the p-Torsion of convex sets. In particular, a Pólya-type inequality in any dimension and two quantitative estimates in dimension two are proved.

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