Program

	Thursday, February 22 Accademia SLA Corso Vittorio Emanuele, 59		Friday, February 23 Aula L1.3 Dipartimento FIM Via Campi 213/A		Saturday, February 24 Accademia SLA Corso Vittorio Emanuele, 59
		09.00–09.40	Giulia Treu	09.00–09.40	Francesco Leonetti
		09.50–10.30	Stefano Pagliarani	09.50–10.30	Vittorio Martino
		10.30–11.00	coffee break	10.30–11.00	coffee break
		11.00–11.40	Flavia Giannetti	11.00–11.40	Fabiana Leoni
		11.50–12.30	Francesco Della Pietra	11.50–12.30	Giovanni Cupini
14.00–14.40	Andrea Cianchi	14.30–15.10	Alessia Kogoj
14.50–15.30	Isabeau Birindelli	15.20–15.40	Mirco Piccinini
15.40–16.00	Pasquale Ambrosio	15.40–16.00	Andrea Gentile
16.00–16.30	coffee break	16.00–16.30	coffee break
16.30–16.50	Antonio Giuseppe Grimaldi	16.30–16.50	Francesca Anceschi
16.50–17.10	Annalaura Rebucci	16.50–17.10	Gloria Paoli
17.20–17.40	Giulio Pecorella	17.20–17.40	Andrea Torricelli
17.40–18.00	Giacomo Bertazzoni	17.40–18.00	Erica Ipocoana

Thursday, February 22
14.00–14.40	Andrea Cianchi (Università degli Studi di Firenze) Title: Local boundedness of minimizers under unbalanced Orlicz growth conditions Abstract: Local minimizers of integral functionals of the calculus of variations are analyzed under growth conditions dictated by different lower and upper bounds for the integrand. Growths of non-necessarily power type are allowed. The local boundedness of the relevant minimizers is established under a suitable balance between the lower and the upper bounds. Classical minimizers, as well as quasi-minimizers are included in our discussion. Functionals subject to so-called $p,q$-growth conditions are embraced as special cases and the corresponding sharp results available in the literature are recovered. This is a joint work with Mathias Schaeffner.
14.50–15.30	Isabeau Birindelli (Sapienza Università di Roma) Title: Some degenerate non local operators: regularity and qualitative properties Abstract: In collaboration with Giulio Galise, Hitoshi Ishii, Erwin Topp, Yannick Sire, we have studied non local operators whose nonlocality is only along some directions and which are non linear and degenerate in the sense that they approximate degenerate elliptic local operators. In this talk we will concentrate on the Hölder regularity of the solutions.
15.40–16.00	Pasquale Ambrosio (Università degli Studi di Napoli Federico II) Title: Regularity results for weak solutions to widely degenerate parabolic problems Abstract: The aim of the talk is to show some recent higher differentiability results for solutions to degenerate parabolic PDEs whose principal part behaves like a $p$-Laplace operator only outside a ball [1,3]. Next, I will present some new gradient bounds for strongly singular or degenerate parabolic systems with data in Orlicz spaces [2]. The talk will also offer the opportunity to describe the state of the art on similar topics. References [1] P. Ambrosio, Fractional Sobolev regularity for solutions to a strongly degenerate parabolic equation, Forum Mathematicum (2023). [2] P. Ambrosio, F, Bäuerlein, Gradient bounds for strongly singular or degenerate parabolic systems, preprint (2023), arXiv. [3] P. Ambrosio, A. Passarelli di Napoli, Regularity results for a class of widely degenerate parabolic equations, Adv. Calc. Var. (2023).
16.00–16.30	coffee break
16.30–16.50	Antonio Giuseppe Grimaldi (Università degli Studi di Napoli Federico II) Title: Gradient regularity for very degenerate convex integrals Abstract: We present a regularity result for minimizers of integral functionals of the Calculus of Variations of the form $$ \mathcal{F}(u) := \int_\Omega \dfrac{1}{p} \left(|Du(x)|_{\gamma(x)}-1 \right)_+^{p} \ \mathrm{d}x , $$ where $p >1$, $u : \Omega \subset \mathbb{R}^n \to \mathbb{R}^N$, with $N \ge 1$, is a possibly vector-valued function, and $| \cdot |_\gamma$ is the associated norm of a bounded, symmetric and coercive bilinear form on $\mathbb{R}^{Nn}$. We show that $\mathcal{K}(x,Du)$ is continuous in $\Omega$, for any continuous function $\mathcal{K}: \Omega \times \mathbb{R}^{Nn} \rightarrow \mathbb{R}$ vanishing on $\big\{ (x,\xi ) \in \Omega \times \mathbb{R}^{Nn} : |\xi|_{\gamma(x)} \le 1 \big\}$. The main feature of functionals as above is that they are widely degenerate convex. In the case $\gamma= \rm{Id}$, these functionals degenerate on the unit ball centered at the origin and naturally arise as a model for optimal transport problems with congestion effects. Here, we deal with functionals with a more general degeneracy set.
16.50–17.10	Annalaura Rebucci (Max Planck Institute – Leipzig) Title: On the De Giorgi-Nash-Moser regularity theory for hypoelliptic operators Abstract: We discuss recent results that lie at the intersection of two major lines of research in the theory of partial differential equations. The celebrated De Giorgi–Nash–Moser theorem provides Hölder estimates and the Harnack inequality for solutions to divergence form elliptic or parabolic equations with rough coefficients. Hörmander’s theory of hypoellipticity provides geometric conditions for the regularity of solutions to operators when ellipticity fails in some directions. We here present new results that extend the De Giorgi-Nash-Moser theory to a large class of hypoelliptic operators with rough coefficients. A key ingredient to prove these results is a Poincaré inequality, which we derive from the construction of suitable trajectories. The trajectories we rely on are quite flexible and allow us to consider equations with an arbitrary number of commutators and whose diffusive part is either local (second-order) or non-local (fractional order). We later combine the Poincaré inequality with a $L^2–L^\infty$ estimate, a Log-transformation and a classical covering argument (Ink-Spots Theorem) to deduce Harnack inequalities and Hölder regularity along the line of De Giorgi method.
17.20–17.40	Giulio Pecorella (Università degli Studi di Modena e Reggio Emilia) Title: A study of the Kuramoto model for synchronization phenomena based on degenerate Kolmogorov-Fokker-Planck equations Abstract: We consider a nonlinear partial differential equation that arises in a generalization of the Kuramoto model. Based on the known theory of Kolmogorov operators, we prove existence, uniqueness and a priori estimates of the solution to a Cauchy problem which describes the synchronization with inertia of a continuum of oscillators. Some numerical experiments are shown in order to validate the Kuramoto model with inertia. This is a joint work in collaboration with S. Polidoro and C. Vernia.
17.40–18.00	Giacomo Bertazzoni (Università degli Studi di Modena e Reggio Emilia) Title: Convergence of  generalized Orlicz norms with lower growth rate tending to infinity Abstract: In this work, in collaboration with Petteri Harjulehto and Peter Hästö, we study convergence of generalized Orlicz energies when the lower growth-rate tends to infinity. We generalize results by Bocea–Mihăilescu (Orlicz case) and Eleuteri–Prinari (variable exponent case) and allow weaker assumptions: we are also able to handle unbounded domains with irregular boundary and non-doubling energies.

Friday, February 23
09.00–09.40	Giulia Treu (Università degli Studi di Padova) Title: Local Lipschitz continuity for energy integrals with slow growth and lower order terms Abstract: We consider integral functionals with slow growth and explicit dependence on $u$ of the lagrangian; this includes many relevant examples, as, for instance, in elastoplastic torsion problems or in image restoration problems. In a joint paper with Michela Eleuteri and Stefania Perrota (Università di Modena e Reggio Emilia) we prove that the local minimizers are locally Lipschitz continuous. The proof makes use of recent results obtained in collaboration with Flavia Giannetti (Università di Napoli Federico II) concerning the Bounded Slope Condition.
09.50–10.30	Stefano Pagliarani (Alma Mater Studiorum – Università di Bologna) Title: Degenerate McKean-Vlasov equations with singular dirft Abstract: We study a class of McKean-Vlasov stochastic differential equations (MKV SDEs) with degenerate diffusion, a kinetic Langevin-type model being a particular instance. The MKV interaction acts on the drift through the multiplication between the density of the solution and a distribution that belongs to suitable anisotropic Besov space. These equations can be understood as mean-field limits of particle systems with singular moderate interactions. We prove well-posedness for the non-linear singular martingale problem associated to the MKV SDE and obtain partial regularity results for the density of the time-marginals. The approach combines analytical and probabilistic tools. As a by-product, we obtain well-posedness and stability results for the relevant non-linear singular Fokker-Planck and singular Kolmogorov backward PDEs, and the well-posedness of the linear kinetic-type singular martingale problem. This is a joint work with Elena Issoglio, Francesco Russo and Davide Trevisani.
10.30–11.00	coffee break
11.00–11.40	Flavia Giannetti (Università degli Studi di Napoli Federico II) Title: Higher Differentiability for Solutions of Stationary Navier-Stokes Systems Abstract: We consider stationary systems of the type \begin{equation} \begin{split} & – {\rm div } a(x, \mathcal{E} u)+\nabla\pi+B(x,u,Du)=f \\ & {\rm div } u=0 \end{split} \end{equation} in $\Omega\subset \mathbb{R}^n$ bounded open set, where $\mathcal{E} u=\frac{1}{2}(Du+Du^T)$ is the symmetric part of the gradient. Assuming the function $a(x,\xi)$ depending Hölder continuously on $x$ and satisfying, with respect to $\xi$, either $p$-growth conditions with $p\ge 2$ and $p>\frac{3n}{n+2}$, or $\phi$-growth conditions, with $\phi$ appropriate Young function, we study the fractional higher differentiability of weak solutions $(u,\pi)$ to the above system. The results that will be illustrated have been obtained in two recent papers in collaboration with A.Passarelli di Napoli and C. Scheven.
11.50–12.30	Francesco Della Pietra (Università degli Studi di Napoli Federico II) Title: Optimal estimates for the first Robin eigenvalue of the $p$-Laplacian Abstract: In this talk, I will discuss some optimal upper and lower bounds for the first Robin eigenvalue of the $p$-Laplacian operator, with Robin boundary conditions. These bounds are expressed in terms of geometrical quantities related to the domain, such as the volume, perimeter and inradius. I will also discuss the case where $p$ approaches 1.
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14.30–15.10	Alessia Kogoj (Università degli Studi di Urbino Carlo Bo) Title: Subelliptic Liouville Theorems  Abstract: Several Liouville-type theorems are presented, related to evolution equations and to their “time”-stationary counterpart. The equations we are dealing with are left translations invariant on a Lie group structure and, in some cases, homogeneous with respect to a group of dilations. In all these cases the operators have smooth coefficients and are hypoelliptic.  We also present a “polynomial” Liouville-type theorem for $X$-elliptic operators with nonsmooth coefficients, by extending to this new setting a celebrated result by Colding and Minicozzi related to the Laplace-Beltrami operator on Riemannian manifolds. The results are contained in a series of papers in collaboration with A. Bonfiglioli, E. Lanconelli, Y. Pinchover, S. Polidoro and E. Priola.
15.20–15.40	Mirco Piccinini (Università di Pisa) Title: On the lack of compactness in the critical Sobolev embedding in the Heisenberg group Abstract: In the sub-Riemannian setting of the Heisenberg group, we will investigate the effects of the lack of compactness in the critical Sobolev embedding. In particular, by means of variational techniques, we will show that optimal functions of a subcritical energy approximation of the Sobolev quotient in bounded (not necessarily regular) domains do concentrate energy at a single point. Moreover, assuming extra geometric features on the involved domain, in line with the particular underlying geometric framework, we show that the concentration point can be localized by the Green’s function, thus proving that a conjecture of Brezis and Peletier (Essays in honor of Ennio De Giorgi 1989) does hold in the Heisenberg group.
15.40–16.00	Andrea Gentile (Scuola Superiore Meridionale) Title: A quantitative result in spectral theory Abstract: Shape Optimization deals with the maximization/minimization of functional depending on domain. When one knows the optimal shape (the maximum/minimum of the shape optimization problem), an interesting question arises: can we say something when a domain almost achieves the maximum/minimum value? The answer to this question is a quantitative inequality, i.e. an inequality stating that if the value of functional is close to the maximum, then the domain is “close” (in some sense) to the optimal shape. In this work we present a quantitative result in spectral theory: more precisely we get an improvement of the result contained in [HM22; BNT16; Krö98] obtaining a quantitative inequality for the Neumann-Laplacian on convex sets in terms of width and diameter. References [BNT16] L. Brasco, C. Nitsch, and C. Trombetti. “An inequality à la Szegö-Weinberger for the $p$-Laplacian on convex sets”. In: Commun. Contemp. Math. 18.6 (2016), pp. 1550086,23. [HM22] A. Henrot and M. Michetti. “Optimal bounds for Neumann eigenvalues in terms of the diameter”. In: (2022). [Krö98] P. Kröger. “On the ranges of eigenfunctions on compact manifolds”. In: Bull. London Math. Soc. 30.6 (1998), pp. 651–655.
16.00–16.30	coffee break
16.30–16.50	Francesca Anceschi (Università Politecnica delle Marche) Title: Cauchy problems for linear and nonlinear models in kinetic theory Abstract: In this talk, I will discuss some recent existence results for Cauchy problems associated with linear and nonlinear models in kinetic theory. The results I present are mainly developed in two joint works, one with Y. Zhu, and the other in collaboration with S. Muzzioli and S. Polidoro.
16.50–17.10	Gloria Paoli (Università degli Studi di Napoli Federico II) Title: The Talenti comparison result in a quantitative form Abstract: We obtain a quantitative version of the classical comparison result of Talenti for elliptic problems with Dirichlet boundary conditions. The key role is played by quantitative versions of the Pólya-Szegő inequality and of the Hardy-Littlewood inequality. This is a joint work in collaboration with Vincenzo Amato, Rosa Barbato and Alba Lia Masiello.
17.20–17.40	Andrea Torricelli (Politecnico di Torino) Title: Variationals Inequalities for Obstacle Problems with General Growth Abstract: Variational inequalities play a foundamental role when studing the Sobolev regularity of solutions to the obstacle problem. It is well known that when the integrad satisfies $p$-growth conditions or $(p,q)$-growth conditions then it is possible to prove that the solutions to the obstacle problem satisfy a related variational inequality. In this talk I show how to prove the aforementioned variational inequality when working with more general growths.
17.40–18.00	Erica Ipocoana (Freie Universität Berlin) Title: On a phase-field model for tumor growth Abstract: The study of tumor growth processes has become of great interest also for mathematicians in recent years. Indeed, mathematical models might be able to give further insights in tumor growth, focusing on the reality-matching mathematical assumptions of parameters and analysis of long-time behaviour. In particular, the framework of diffuse interface modeling has received increasing attention. In this context, the tumor is seen as an expanding mass surrounded by healthy tissues, while the interface in between contains a mixture of both healthy and tumor cells. Our aim is to derive a phase-field model describing the growth of a tumor, whose evolution is assumed to be governed by biological mechanisms such as proliferation of cells via nutrient consumption and apoptosis. More precisely, we model the process through a non-isothermal Allen-Cahn system following the approach based on microforce balance and then study its well-posedness. Namely, we are able to prove the existence and uniqueness of a local solution to our model by Galerkin’s approach.

Saturday, February 24
09.00–09.40	Francesco Leonetti (Università degli Studi dell’Aquila) Title: Elliptic systems and double phase functionals Abstract: It is well known that solutions to elliptic systems may be unbounded. Nevertheless, for some special classes of systems, it can be proved that solutions are bounded. We mention a recent result of this kind and we discuss some examples suggested by double phase functionals.
09.50–10.30	Vittorio Martino (Alma Mater Studiorum – Università di Bologna) Title: Compactness results for the Dirac-Einstein functional Abstract: Starting from motivations coming from Physics, we will introduce the Dirac-Einstein functional on a spin manifold and we will show the compactness of the variational solutions. Moreover, after restricting the functional on a conformal class, we will show the classification of the Palais-Smale sequences and some results regarding the existence of solutions for the related Dirac-Einstein equations.
10.30–11.00	coffee break
11.00–11.40	Fabiana Leoni (SAPIENZA Università di Roma) Title: Fully nonlinear equations with singular terms in punctured balls Abstract: We consider radial solutions of fully nonlinear, uniformly elliptic equations, posed in punctured balls, in presence of radial singular quadratic potentials. We discuss both the principal eigenvalues problem, obtaining an extension in the fully nonlinear framework of the Hardy-Sobolev constant, and the classification of solutions based on the asymptotic behavior near the singularity, for equations having also superlinear zero order  terms.
11.50–12.30	Giovanni Cupini (Alma Mater Studiorum – Università di Bologna) Title: The Leray-Lions existence theorem under $p,q$-growth conditions Abstract: In this talk I will describe recent results obtained in collaboration with P. Marcellini and E. Mascolo. We first proved regularity results (local boundedness, Lipschitz continuity and higher differentiability) of weak solutions to second order elliptic equations in divergence form with $(x,u)$-dependence other than on the gradient variable and satisfying $p,q$-growth conditions. These results were then used to prove an existence result of weak solutions to a Dirichlet problem associated to these equations. This is a first attempt to extend to $p,q$-growth the well known Leray-Lions existence theorem, which holds under the so-called natural growth conditions.