Calculus of Variations and Geometric Measure Theory

Semilinear eigenvalue problems for the Dirichlet Laplacian

Giovanni Franzina (IAC/CNR)

created by gelli on 01 Apr 2019

10 apr 2019 -- 17:00   [open in google calendar]

Sala Seminari (Dipartimento di Matematica di Pisa)


The talk overviews the qualitative properties of critical points and critical values of Dirichlet integrals on open sets of $\mathbb R^N$, with homogeneous Dirichlet boundary conditions, subject to $L^q$ constraints with $q$ ranging between $1$ and the Sobolev exponent $2N/(N-2)$. The associated eigenvalue-type problem is mildly non-local, except in the case of Helmoltz equation ($q=2$). We discuss to which extent properties typical of the linear eigenvalues, such as the simplicity of the first eigenvalue on connected sets, its isolation, and the discreteness of the spectrum, hold true if either $q<2$ or $q>2$, presenting both some positive results and some counterexamples. This is a joint work with Lorenzo Brasco (Università di Ferrara)