Speakers

In this talk, we will discuss the regularity of solutions to stationary and evolutionary equations driven by infinite dimensional second order Kolmogorov operators with unbounded drift. Moreover, we will analyze the connection between control theory and the regularity of Kolmogorov operators. Finally, we will examine how these results apply to the theory of regularization by noise for stochastic partial differential equations.

We discuss a nonlocal fractional problem that serves as a nonlocal counterpart of the classical problem of functions of least gradient. We show how we obtain the existence of minimizers by using their connection to nonlocal minimal sets. Additionally, we discuss the continuity of these minimizers and a weak formulation of the problem. The results presented are obtained in collaboration with S. Dipierro, L. Lombardini, J. Mazón and E. Valdinoci.

The Newtonian potential of a merely continuous function $f$ has not, in general, continuous second-order derivatives. As a consequence, the Poisson equation $\Delta u = - f$ is not, in general, solvable in the pointwise classical sense. However, a 1909 result by Pizzetti shows that the previous Poisson equation is pointwise solvable in a suitable generalized sense for every continuous function $f$. This Pizzetti’s result is based on the so called Poisson-Jensen mean value formula for classical solutions to the Poisson equation. In this talk we show how Pizzetti's approach also works for wide classes of hypoelliptic linear second order PDEs.

In this talk, I will introduce the conformal fractional Dirac operator, which is the spinorial counterpart of the classical conformal fractional Laplacian. Moreover, I will present the Caffarelli-Silvestre type extension that allows us to define this fractional operator as a Dirichlet-to-Neumann type operator. As a result, we get several energy identities for the conformal fractional Dirac operator, along with a spinorial weighted Sobolev type identity. In the second part of the talk, I will focus on the critical operator and the corresponding spinorial Q-curvature.

In this talk, I will discuss existence, classification, and nondegeneracy results for solutions to singular Liouville-type equations in one dimension. The problem arises in the mathematical modeling of galvanic corrosion phenomena in ideal electrochemical cells, where an electrolyte solution is confined within a bounded domain with an electrochemically active portion of the boundary. In higher dimensions, Liouville equations appear in prescribed curvature problems in conformal geometry: solutions correspond to metrics with constant Q-curvature on Euclidean space, with a singular point at the origin. After a general overview of the existing literature, I will focus on the one-dimensional case and show that solutions are nondegenerate under mild assumptions on the singular weight. The proof relies on harmonic extensions and conformal transformations, which allow us to rewrite the linearized Liouville equation as a Steklov eigenvalue problem on either the intersection or the union of two disks. These results were obtained in collaboration with A. DelaTorre, A. Pistoia, and L. Provenzano.

In this talk I will briefly present some recent results concerning nonexistence of nontrivial nonnegative solutions to semilinear elliptic and parabolic inequalities with a potential on infinite weighted graphs. Given a distance on the graph, we assume an upper bound on its Laplacian and a growth condition on a suitably weighted volume of balls. Under such hypotheses, we prove that the problem does not admit any nonnegative nontrivial solution. We will see some examples, that show that our conditions are optimal. This is based on joint works with F. Punzo and J. Somaglia (Politecnico di Milano).

An Orlicz space approach to exponential elliptic problems in higher dimensions

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We will present some recent results on the first eigenvalue of the magnetic Laplacian associated with closed potential 1-forms on compact surfaces. We introduce some (conformal) spectral invariants, naturally associated with the first eigenvalue, which can be estimated by the geometry of the Jacobian torus of the surface. In genus 1 the computations are explicit, and one can have a simple geometric interpretation of the invariants. Then we define a notion of magnetic isospectrality and we prove that if two Riemannian metrics are magnetic-isospectral, then they are conformal and have the same volume. Joint work with Bruno Colbois (Université de Neuchâtel) and Alessandro Savo (Sapienza Università di Roma)