Bruno Pini Mathematical Analysis Seminar

On the De Giorgi-Nash-Moser regularity theory for kinetic equations

Francesca Anceschi — 2024-01-09

In this note we review some recent results regarding the De Giorgi-Nash-Moser weak regularity theory for Kolmogorov operators obtained in [10] in collaboration with A. Rebucci. To simplify the treatment, we focus on the model case of the Fokker-Planck equation with rough coefficients and we highlight the main steps of the proof of a Harnack inequality for weak solutions.

Regularity results for isoperimetric sets with density

Eleonora Cinti — 2024-01-09

In this note, we present some recent regularity results for sets which minimize a weighted notion of perimeter under a weighted volume constraint. We focus on the case of two different densities which are merely alpha-Holder continuous, and describe what are the main issues and techniques used in order to establish the optimal regularity C^{1, alpha/(2-alpha)}for the reduced boundary of such sets.

A density result on a BV-type space on Carnot groups

Annalisa Baldi — 2024-01-09

In the setting of Carnot groups (connected, simply connected and stratified Lie groups), we prove a density result for a BV-type space previously introduced in [3]. In addition, we relate the dual of this BV-type space with the dual of the well known space of functions of intrinsic bounded variation. These results extend to the setting of Carnot groups some properties studied by Phuc e Torres in [22] and [23] in the Euclidean setting.

A brief note on Harnack-type estimates for singular parabolic nonlinear operators

Eurica Henriques — 2024-01-09

In this brief note we introduce Harnack-type inequalities, which are typical in the context of singular nonlinear parabolic operators, and describe their state of art in the context of anisotropic operators.

Some topics on the regularity of analytic-Gevrey vectors

Makhlouf Derridj — 2024-01-09

My aim is to give, in this talk, some topics on the question of regularity of Analytic-Gevrey vectors of partial differential operators (p.d.o.) with analytic-Gevrey coefficients. Since the results obtained in the sixties on elliptic p.d.o's, which are both hypoelliptic (C^∞ setting), analytic-Gevrey hypoelliptic (analytic-Gevrey setting) and satisfy the so-called Kotake-Narasimhan property, a lot of works and articles were devoted to these problems in case of non elliptic p.d.o's under suitable hypotheses (for example on the degeneracy of ellipticity). I will consider the third problem on analytic-Gevrey vectors in the three cases of global (on compact manifolds), local (near a point in the base-space), microlocal (near a point in the cotangent space), situations, and say few words on the main two methods used in order to obtain positive (or negative) results. Finally I will focus on some new microlocal results on degenerate elliptic (also called sub-elliptic) p.d.o's of second order, obtained in a common work with Gregorio Chinni.

Some advances in analytic hypoellipticity

Marco Mughetti — 2024-01-09

We present a brief survey on the theory of the real analytic regularity for the solutions to sums of squares of vector fields satisfying the Hörmander condition.

On the surface average for harmonic functions: a stability inequality

Giovanni Cupini — 2024-01-09

In this article we present some of the main aspects and the most recent results related to the following question: If the surface mean integral of every harmonic function on the boundary of an open set D is "almost'' equal to the value of these functions at x₀ in D, then is D "almost'' a ball with center x₀? This is the stability counterpart of the rigidity question (the statement above, without the two "almost'') for which several positive answers are known in literature. A positive answer to the stability problem has been given in a paper by Preiss and Toro, by assuming a condition that turns out to be sufficient for ∂D to be geometrically close to a sphere. This condition, however, is not necessary, even for small Lipschitz perturbations of smooth domains, as shown in our recent paper, in which a stability inequality is obtained by assuming only a local regularity property of the boundary of D in at least one of its points closest to x₀.

Regularity results for Kolmogorov equations based on a blow-up argument

Annalaura Rebucci — 2024-01-09

We present recent results regarding the regularity theory for degenerate second order differential operators of Kolmogorov-type. In particular, we focus on Schauder estimates for classical solutions to Kolmogorov equations in non-divergence form with Dini-continuous coefficients obtained in [30] in collaboration with S. Polidoro and B. Stroffolini. Furthermore, we discuss new pointwise regularity results and a Taylor-type expansion up to second order with estimate of the rest in L^p norm, following the recent paper [14] in collaboration with E. Ipocoana. The proofs of both results are based on a blow-up technique.

Nonlinear fractional equations in the Heisenberg group

Giampiero Palatucci — 2024-01-09

We deal with a wide class of nonlinear nonlocal equations led by integro-differential operators of order (s,p), with summability exponent p in (1,∞) and differentiability order s in (0,1), whose prototype is the fractional subLaplacian in the Heisenberg group. We present very recent boundedness and regularity estimates (up to the boundary) for the involved weak solutions, and we introduce the nonlocal counterpart of the Perron Method in the Heisenberg group, by recalling some results on the fractional obstacle problem. Throughout the paper we also list various related open problems.

Regularity and Semialgebraicity of Solutions of Linear Equations Systems

Marcello Malagutti — 2024-01-09

This work is concerned with the study of a necessary and sufficient condition for the existence of solutions with a given regularity to a system of linear equations with coefficients of given regularity. First, to properly contextualize the subject matter and to introduce crucial analytical solving tools, we go through results by C.Fefferman- J.Kollár and by C.Fefferman - G.K.Luli.
Then we prove our result to determine a necessary and sufficient condition for the existence of continuous (C⁰) semialgebraic
solutions in case of a system of linear equations with continuous semialgebraic coefficients.

Some results on the Dirac-Einstein equations

Vittorio Martino — 2024-01-09

In this note we introduce the Dirac-Einstein equations on a spin manifold and we review some recent results, in particular: the compactness of the variational solutions, the classification of the Palais-Smale sequences for the related conformal problem, and finally some existence results.

On the regularity of Anisotropic p-Laplacean Operators: the pursuit of a comprehensive theory of regularity

Bashayer Majrashi — 2024-01-09

With this note, we aim at drawing a short, coherent, and compact guide of the state-of-the-art on the theory of basic regularity, such as local boundedness, local Hölder continuity, Harnack estimates and some of their consequences, in the context of solutions to anisotropic p-Laplacean operators, elliptic and parabolic.

Second order p-evolution equations with critical nonlinearity

Marcello D'Abbicco — 2024-01-09

In this paper, we study critical nonlinearities for global small data solutions to the plate equation and other second order p-evolution equations, possibly under the action of a noneffective dissipative term.