Sobolev inequalities in the Alps

Sobolev inequalities in the Alps 2023

29-30 Jun 2023 Grenoble (France)

Schedule

Talks will be 50 minutes + questions. The conference will be held in room B29, on the ground floor of the Insitut Fourier (turn right from the main entrance, it is the first room on the left after the library).

Thursday 29th of June

12:00-14:00 welcome lunch at Martin's Café for those already in Grenoble

14:00-15:00 Gabriele Grillo

15:00-16:00 Daniele Semola

16:00-16:30 coffee break

16:30-17:30 Baptiste Huguet

17:30-18:30 Ilaria Mondello

20:00 evening diner at L'Epicurien

Friday 30th of June

09:00-10:00 Frédéric Robert

10:00-10:30 coffee break

10:30-11:30 Alberto Farina

11:30-12:30 Giona Veronelli

12:30-14:00 goodbyes and lunch at Martin's Café

How to get there? And other practical information

All the talks will take place at the Institut Fourier, located on the campus of the university, 200m from the tramway line B stop "Bibliotheque Universitaire". Note that tramway B is direct to the train station. Detailed instructions for coming to the Institut can be found here: http://www-fourier.ujf-grenoble.fr/?q=fr/content/contact-et-acces

On the ground floor in front of the entrance of the Institut, you will find the research library which you can freely access (opening hours approximately 08:30-17:00).

On the second floor of the Institut, at the very end of the corridor on the left you will find the cafeteria with comfortable seats, blackboard and coffee (which you can take freely as much as you need, just mark it on the paper next to the coffee machine, under the name "Invités")

In the right aisle of the Institute, you will find teaching classrooms, most of which should be empty, in case you need a blackboard to work.

There have been rare cases of theft in the building, so just to be on the safe side, do not leave your valuables unattended.

Abstracts

Giulio Ciraolo Canceled due to personal reasons

Title: Classification and nonexistence results for critical $p$-Laplace equations

Abstract: We classify positive solutions to a class of critical quasilinear equations with Neumann boundary conditions in convex domains. When the domain is an unbounded cone, we classify the extremals of the Sobolev inequality. In the case of bounded domains, we show the nonexistence of positive solutions and show its relation with critical $p$-Laplace equations. Some results are also generalized to equations arising from Caffarelli-Kohn-Nnirenberg inequalities.

Alberto Farina

Title: Some rigidity results for minimal graphs over unbounded Euclidean domains

Abstract: In the spirit of Bernstein's well-known theorems, I will present some new classification results for minimal Cartesian graphs over unbounded Euclidean domains. In particular, I will show that a positive minimal graph over an open affine half-space, and under the homogenous Dirichlet boundary condition, must be an affine function.

Pdf file of the talk: Alberto's talk

Gabriele Grillo

Title: Fractional nonlinear diffusions on manifolds: well-posedness and smoothing effects.

Abstract: We consider classes of nonlinear diffusions, involving the fractional Laplacian on noncompact manifolds M. The manifold is assumed to support Faber-Krahn (or Nash, or Sobolev) inequalities and, for some of the results, stricter assumptions like being Cartan-Hadamard, namely that M is simply connected with nonpositive curvature, or to satisfy sec< -c < 0, sec denoting sectional curvature. Under the sole assumption that the Faber-Krahn inequality holds on M, we prove well-posedness of the problem, in the weak dual sense, for data belonging to a weighted L1 space where the weight is, roughly speaking, the fractional Green function on the manifold, assumed to exist. Smoothing effects for the evolution are then proved, with stronger results as soon as further assumptions are required. The results stem from joint works with E. Berchio, M. Bonforte, M. Muratori.

Pdf file of the talk: Gabriele's talk

Baptiste Huguet

Title: Weighted functional inequalities for generalised Cauchy measures.

Abstract: Generalized Cauchy measures is a family of measures which does not satisfy classical functional inequalities but only some weighted version of it. The formalism of curvature-dimension criterion allows to unify the study of classical and weighted inequalities. In this talk, I will present how curvature-dimension criterion can provide simple proofs of these inequalities with optimal constant, sharp estimate of the deficit and extremal functions when they exists.

Pdf file of the talk: Baptiste's talk

Ilaria Mondello

Title: Gromov-Hausdorff limits of manifolds with a Kato bound on the Ricci curvature.

Abstract: The goal of this talk is to present some recent results on manifolds for which the negative part of the Ricci curvature satisfies a Kato bound, inspired by Kato potentials in the Euclidean space. This condition is implied for instance by a lower Ricci curvature bound, or an integral Ricci bound in the spirit of Gallot and Petersen-Wei. We will explain some analytic consequences of a Kato bound, and how we used them to study the structure of Gromov-Hausdorff limits of manifolds satisfying this kind of bounds. This talk is based on a joint work with G. Carron and D. Tewodrose.

Frédéric Robert

Title: Localization of peaks for high-order equations.

Abstract: We analyze families (u_i) of solutions to nonlinear polyharmonic equations like $P_iu_i=(-\Delta)^k u_i+…=|u_i|^{4k/(n-2k)}u_$ on a Riemannian manifold $(M^n,g)$ where $n>2k$. We prove that families that behave like u_i=u_\infty+Peak+o(1) concentrate where the limiting operator $P_\infty$ touches (in some sense) the geometric GJMS operator. As a byproduct, we obtain nonexistence of extremals for some high-order Sobolev inequalities.

Pdf file of the talk: Frédéric's talk

Daniele Semola

Title: Isoperimetry and lower Ricci curvature bounds.

Abstract: There is a well-known connection between curvature and isoperimetric inequalities on smooth Riemannian manifolds, motivated in particular by the presence of a Ricci curvature term in the second variation formula for the area of hypersurfaces. The goal of this talk will be to review some recent developments around this classical topic and to illustrate the role of geometric analysis on non-smooth metric measure spaces.

Pdf file of the talk: Daniele's talk

Giona Veronelli

Title: The L^p-positivity preservation on Riemannian manifolds.

Abstract: A Riemannian manifold is said to be L^p-positivity preserving if any L^p (distributional) solution of - Δu +u ≥ 0 is necessarily nonnegative. This property stems from the work of T. Kato and is motivated by, and has applications to, the spectral theory of Schrödinger operators with singular potentials. We will first show that complete smooth manifolds are always L^p-positivity preserving for finite p>1, solving in particular a conjecture by Braverman, Milatovic and Shubin. Then we will move some steps towards both incomplete manifolds and nonsmooth metric spaces.

This is based on joint works with B. Guneysu, S. Pigola, P. Stollmann and D. Valtorta.

Pdf file of the talk: Giona's talk

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