Abstract:
The Gaussian-like p-Laplacian: Dirichlet problem,eigenvalues, and the Faber–Krahn inequality
In this thesis, we extend several classical results for the Euclidean p-Laplacian,
∆pu = div(|∇u|p−2∇u), to a setting where the Lebesgue measure is replaced by a
Gaussian-like measure dμ = Ze−W dLN . Under this measure, the operator takes
the form div(|∇u|p−2∇u) − |∇u|p−2⟨∇u, ∇W ⟩. Our study specifically focuses on
the Dirichlet problem and the associated eigenvalue problem. In particular, for the
weak Dirichlet problem, we prove existence and uniqueness of the solution. As for
the eigenvalue problem, the analysis focuses on the first eigenvalue. We show that
a Faber-Krahn inequality holds in this context as well
