Convegno
“NONPUB24 - NONLOCAL AND NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS AT THE UNIVERSITY OF BOLOGNA”

In this talk, we investigate qualitative properties of singular solutions to some elliptic problems. In the first part, we will focus our attention on semilinear and quasilinear elliptic problems under zero Dirichlet boundary conditions. In the second part, thanks to the previous analysis, we obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

In this talk, we consider n-dimensional Grushin spaces, where a Riemannian metric degenerates along a line in the space, resulting in a sub-Riemannian structure. We discuss the validity of (sub-)mean value property for (sub-)harmonic functions on hypersurfaces within Grushin spaces of dimension n>2. Our interest is driven by the classical counterpart: mean value formulas for harmonic functions on surfaces in the Euclidean setting are crucial for establishing the Bombieri-De Giorgi-Miranda gradient bound, which, in turn, plays a central role in the classical regularity theory. We conclude by presenting remarks and open questions about the regularity theory of minimal surfaces within this sub-Riemannian framework, which is yet to be established.

One of the oldest and most studied problems in the spectral theory of differential operators is the eigenvalue problem for the Dirichlet Laplacian. Classical questions about Laplacian eigenvalues concern their sensitivity analysis, optimization, asymptotic expansions, and many other more properties. On the other hand, similar questions remain open for an important class of degenerate operators, that is the Grushin Laplacians. In this talk I will present some recent results regarding the spectral theory of the Grushin Laplacian and in particular its shape sensitivity analysis, the optimization of the first eigenvalue, and a blow-up analysis.

We look at spreading phenomena under the action of singular potentials modeling repulsion between the liquidgas interface and the substrate. We mainly discuss the static case: depending on the form of the potential, the macroscopic profile of equilibrium configurations can be either droplet-like or pancake-like, with a transition profile between the two at zero spreading coefficient. These results generalize, complete, and give mathematical rigor to de Gennes’ formal discussion of spreading equilibria. Uniqueness and non-uniqueness phenomena are also discussed. Then we will briefly focus on the dynamics, assuming zero slippage at the contact line. Based on formal analysis arguments, we report that generic travelling-wave solutions exist and have finite rate of dissipation, indicating that singular potentials stand as an alternative solution to the contact-line paradox. In agreement with equilibrium configurations, travelling-wave solutions have microscopic contact angle equal to π/2 and, for mild singularities, finite energy. This is a joint work with Lorenzo Giacomelli.

The well-known isocapacitary inequality states that balls minimize the capacity among all sets of the same given volume. In the talk, we prove a sharp quantitative form of this classical result. Namely, we show that the difference between the capacity of a set and that of a ball with the same volume bounds the square of the Fraenkel asymmetry of the set. We then discuss some possible extensions.

The regularity of minimizers of the classical one-phase Bernoulli functional was deeply studied after the pioneering work of Alt and Caffarelli. More recently, the regularity of almost minimizers was investigated as well. We present a regularity result for almost minimizers for a one-phase Bernoulli-type functional in Carnot Groups of step two. Our approach is inspired by the methods introduced by De Silva and Savin in the Euclidean setting. Moreover, some recent intrinsic gradient estimates have been employed. Generalizations to the nonlinear framework will be discussed. Some of the results presented are obtained in collaboration with F. Ferrari (University of Bologna) and N. Forcillo (Michigan State University).