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During this year the seminars were organized by Nuno Carneiro, Leonardo Felicetti, Chiara Gambicchia, Filippo Paiano, Maria Antonietta Palladino, Federico Renzi and Federico Vitillaro.
Regularity under non-uniform ellipticity: known results and recent advances
Regularity under non-uniform ellipticity: known results and recent advances
Gabriele Giannone (University of Firenze)
Gabriele Giannone (University of Firenze)
Aula Fermi - 07 Ottobre 2026
Abstract
Abstract
The question of regularity — whether solutions inherit the smoothness of their underlying equations — is a central, unifying theme in partial differential equations. This talk begins by reviewing the foundational ideas of elliptic regularity, from De Giorgi’s groundbreaking resolution of Hilbert’s XIXth problem to the key distinctions between linear and nonlinear regimes. We then turn to nonuniformly elliptic problems, with a particular focus on equations exhibiting (p,q)-growth, where ellipticity and growth are controlled by distinct powers of the gradient. In this context, regularity is no longer guaranteed: classical counterexamples by Marcellini and Giaquinta demonstrate that minimizers can be unbounded if the gap between p and q is too large. Finally, we discuss recent developments in quite general frameworks, providing a natural setting for analyzing genuinely non-homogeneous problems.
Rank-one properties of BV functions in Carnot groups
Rank-one properties of BV functions in Carnot groups
Alberto Gervani (Scuola Normale Superiore)
Alberto Gervani (Scuola Normale Superiore)
Aula Contini - 21 Ottobre 2026
Abstract
Abstract
The Rank-one theorem is an important structure result for the singular part of distributional gradients of BV functions, first proved by G. Alberti. In this talk, we will first cover two recent proofs of this result (De Philippis-Rindler, Massaccesi-Vittone) and a subsequent generalization of the theorem (for horizontally BV functions) to a particular class of Carnot groups (Don-Massaccesi-Vittone): these stratified Lie groups are the prototypical example of Carnot-Carathéodory spaces in Sub-Riemannian Geometry.
We will then present an adaptation of the work of De Philippis-Rindler to this setting, thereby extending the Rank-one theorem to all Carnot groups. This is achieved via the hypoellipticity of the PDE constraints satisfied by distributional horizontal gradients, for which we will also need a number of results in the theory of Singular Integrals on Homogeneous Spaces.
This is joint work with G. De Philippis, A. Massaccesi, D. Vittone (University of Padova).
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