Seminar in Geometric Function and Mapping Theory

16 November (11:00) - room 403

Ivan Yuri Violo (La Sapienza Università di Roma)

Liouville theorems and a-priori estimates for elliptic equations on Riemannian manifolds

A Liouville-type theorem typically states that bounded global solutions to certain PDEs must be constant. A classical strategy for proving such results is to establish suitable a priori estimates on balls and then let their radius tend to infinity. In Euclidean space, a converse principle is also available: local a priori estimates can often be derived from Liouville theorems by means of a suitable rescaling and compactness argument. In this talk, we will explore this strategy in the setting of Riemannian manifolds and use it to derive local estimates for a class of nonlinear elliptic equations. A key feature of the argument is that it naturally leads to the analysis of non-smooth metric spaces, which plays an essential role in establishing the desired estimates.

19 November

Iván Caamaño Aldemunde (Universidad de Granada)

TBA

26 November (14:00)

Marcin Gryszówka (Uniwersytet Warszawski)

TBA

3 December

Giacomo del Nin (University of Pisa)

TBA

10 December

Wojciech Górny (Uniwersytet Warszawski)

TBA

5 November

Julian Weigt (Oulun yliopisto)

Regularity of maximal functions and a Vitali covering lemma for the boundary

The classical Hardy-Littlewood maximal function theorem states that the Hardy-Littlewood maximal operator is a bounded operator on Lp(Rd) if and  only if 1 < p ≤ ∞. The main ingredient to the proof is the Vitali covering lemma, which roughly states that any bounded set of balls has a pairwise disjoint subset which represents the volume of the original set well.

In 1997 Juha Kinnunen proved the corresponding result for the gradient of the maximal function, i.e. that the Lp(Rd)-norm of the gradient of the maximal function is controlled by the  Lp(Rd)-norm of the gradient of the function if 1 < p ≤ ∞. However, he provides no counterexample in the endpoint p = 1, and so in 2004 Hajłasz and Onninen formally posed the question if the endpoint gradient bound also holds.

We present the current state of research on the problem and show a new variant of the Vitali covering lemma that concerns the perimeter instead of the volume.

22 October

Fernando Ballesta Yagüe (Universidad Complutense de Madrid)

Motivating weighted theory problems through boundary value problems

In this talk, we will explain how boundary value problems for the Laplacian in planar Lipschitz graph domains can be studied through weighted theory. We will consider boundary data satisfying certain integrability conditions, like  Lp  or  Lp,1 , and we will impose different boundary conditions, like Dirichlet, Neumann or a directional derivative.

The technique, introduced by Carlos Kenig in the 80’s, uses a conformal mapping to transfer the problem to the upper half-plane. There, the problem can be treated using Harmonic Analysis techniques. We will emphasise what kind of weighted theory problems are motivated by this approach.

This is joint work with María J. Carro and Virginia Naibo.

15 October

Tommaso Bertin (IMPAN)

Lavrentiev Phenomenon and integral representation of the lower semi-continuous envelope for functionals with non convex and non continuous Lagrangians

In this seminar, we investigate classical integral functionals in the Calculus of Variations and study conditions on the Lagrangian that guarantee the absence of the Lavrentiev Phenomenon. In particular, we focus on Lagrangians that are non-convex, possibly highly discontinuous, and unbounded with respect to the gradient variable.

Our first step to prove the non-occurrence of the Lavrentiev Phenomenon is to prove the representation of the lower semicontinuous envelope of a functional defined in W1,∞(Ω) as an integral functional whose Lagrangian is given by the bipolar of the original one.

Specifically, we adapt a technique presented in the classical book by Ekeland and Temam for the bounded case, and a refinement of a method due to Cellina, for the unbounded case. The integral representation also allows us to apply recent results by Bousquet, Mariconda and Treu to the non-convex setting.

We establish the integral representation of the lower semicontinuous envelope and the corresponding absence of the Lavrentiev Phenomenon under very weak assumptions in the autonomous case and under suitable anti-jump conditions on the spatial variable in the non-autonomous case. Furthermore we prove the strong convergence of the approximating sequence for minimizers or under specific growth assumptions on the Lagrangian.

These results provide new insights into the interplay between regularity assumptions, relaxation methods, and the avoidance of the Lavrentiev Phenomenon, thereby extending recent advances to a broader class of variational problems.

This is a joint project with with G. Treu (Università di Padova) and P. Huguet (université de Toulouse III Paul Sabatier).