Barbato, Rosa (2024) Qualitative and geometric aspects of elliptic equations with Dirichlet or Robin boundary conditions. [Tesi di dottorato]

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Abstract

In this thesis work, we focused on the study of qualitative and geometric aspects of some elliptic equations with Dirichlet or Robin boundary conditions. We start with symmetrization techniques, in particular we study the qualitative properties of the solutions of the second-order elliptic problems with Dirichlet or Robin boundary conditions. Then we focus our attention on the study of Talenti's classical comparison result by proving a quantitative version of it. This is the starting point of this thesis work. However, we also focused our attention on nonlinear operators, in particular the p-laplacian operator. We studied eigenvalue problems with Robin boundary condition, in the anisotropic case and in the Euclidean case. More precisely, in the anisotropic case, we studied the asymptotic behavior of the first Robin eigenvalue λ_p (β) when p goes to 1, proving an isoperimetric inequality for its limit. While in the Euclidean case, several problems were tackled, including estimates on the first Robin eigenvalue λ_p (β) for nonlinear elliptic operators, proving an estimate from above, involving the first Dirichlet eigenvalue, and an estimate from below. Finally, the last part of the thesis contains results obtained by considering problems related to thermal insulation.

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