Recent seminars

Europe/Lisbon — Online

Jean Van Schaftingen

Jean Van Schaftingen, Université Catholique de Louvain
Ginzburg-Landau functionals on planar domains for a general compact vacuum manifold (**postponed**)

Ginzburg-Landau type functionals provide a relaxation scheme to construct harmonic maps in the presence of topological obstructions. They arise in superconductivity models, in liquid crystal models (Landau-de Gennes functional) and in the generation of cross-fields in meshing. For a general compact manifold target space we describe the asymptotic number, type and location of singularities that arise in minimizers. We cover in particular the case where the fundamental group of the vacuum manifold in nonabelian and hence the singularities cannot be characterized univocally as elements of the fundamental group.

We obtain similar results for \(p\)-harmonic maps with \(p\) going to \(2\).

The results unify the existing theory and cover new situations and problems.

This is a joint work with Antonin Monteil (Paris-Est Créteil, France), Rémy Rodiac (Paris-€“Saclay, France) and Benoit Van Vaerenbergh (UCLouvain).


(more information soon)

Europe/Lisbon — Online

Kelei Wang

Kelei Wang, Wuhan University
Regularity of transition layers in Allen-Cahn equation

In this talk I will survey the regularity theory for transition layers in singularly perturbed Allen-Cahn equation, from zeroth order regularity to second order one. Some applications of this regularity theory will also be discussed, including De Giorgi conjecture, classification of finite Morse index solutions and construction of minimal hypersurfaces by Allen-Cahn approximation.

Europe/Lisbon — Online

Simone Dovetta

Simone Dovetta, Università degli Studi di Roma "La Sapienza"
Action versus energy ground states in nonlinear Schrödinger equations

The talk investigates the relations between normalized critical points of the nonlinear Schrödinger energy functional and critical points of the corresponding action functional on the associated Nehari manifold.

First, we show that the ground state levels are strongly related by the following duality result: the (negative) energy ground state level is the Legendre—Fenchel transform of the action ground state level. Furthermore, whenever an energy ground state exists at a certain frequency, then all action ground states with that frequency have the same mass and are energy ground states too. We see that the converse is in general false and that the action ground state level may fail to be convex. Next we analyze the differentiability of the ground state action level and we provide an explicit expression involving the mass of action ground states. Finally we show that similar results hold also for local minimizers, and we exhibit examples of domains where our results apply.

The matter of the talk refers to joint works with Enrico Serra and Paolo Tilli.

Europe/Lisbon — Online

Juraj Földes

Juraj Földes, University of Virginia
Stochastic approach to boundary regularity of hypoelliptic PDEs

We will discuss the almost sure behavior of solutions of stochastic differential equations(SDEs) as time goes to zero. Our main general result establishes a functional law of the iterated logarithm (LIL) that applies in the setting of SDEs with degenerate noise satisfying the weak Hormander condition. We will introduce large deviations to provide some details of proofs. Furthermore, we apply the stochastic results to the problem of identifying regular points for hypoelliptic diffusions and obtain criteria for well posedness of degenerate equations.

This is a joint work with David Herzog and Marco Carfagnini