The optimization of the principal eigenvalue of indefinite weighted problems settled in bounded domains arises as a natural task in the study of the survival threshold for a species in population dynamics. We study the minimization of such eigenvalue, associated with Neumann boundary conditions, performing the analysis of the singular limit in case of arbitrarily small favourable region. We show that, in this regime, the favourable region is connected and it concentrates on the boundary of the domain. Though widely expected, these properties are still unknown in the general case. This is a joint research with Dario Mazzoleni and Gianmaria Verzini.

Differential inclusions have proved to be an extensively valuable tool in studying PDEs, with applications ranging from establishing the existence of solutions to constrained variational problems to constructing pathological examples in various classes of problems. The talk aims to briefly review the theory of solvability for differential inclusions outlining the different approaches that can be used and discuss some recent applications. One of the characteristics of the theory is that, when applicable, it generates infinitely many solutions; therefore, the problem of finding a criterion to select among them the more meaningful ones naturally arises. Although there is still no general method to address this problem, we will see that a tailored selection criterion can be successfully applied for some problems.

Harmonic functions and the Laplacian are — without exaggeration — among the most studied objects in Mathematics. Notwithstanding the foregoing, we will in this seminar define and discuss the properties of a new type of Laplace operator that makes sense in the general setting of metric measure spaces, and which circumvents the need for partial derivatives.

We discuss a transmission problem driven by the $p$-Laplace operator, equipped with a natural interface condition. Two aspects of the problem entail genuine difficulties in the analysis. First, it lacks representation formulas. Also, its ellipticity may collapse as the gradient vanishes. Our arguments circumvent those difficulties and lead to new regularity estimates. First, we prove local boundedness for the solutions. Then we establish an integral estimate for the gradient in $BMO$ spaces. The latter implies solutions have a borderline Hölder modulus of continuity.