Schedule

 Étienne Sandier - Bifurcating solitonic vortices in a strip 

We study the bifurcation of solutions to the stationary Gross-Pitaevskii equation in an infinite strip from the 1-dimensional black soliton. The parameter of the bifurcation is the width of the strip.

We show that for each integer k​​ there is a bifurcation occurring at an explicit width dk​​. The bifurcating branch of solutions have exactly k​​ vortices of alternating signs.

These results are consistent with recent numerical simulations and experiments on Bose Einstein condensates. Joint work with Amandine Aftalion and Philippe Gravejat

Bruno Premoselli - Non-existence of extremals for the second conformal eigenvalue of the conformal laplacian in dimensions 3 to 10

​Let (M,g)​​ be a closed manifold of dimension n ≥ 3​​. We define the conformal Laplacian of g​​ as follows:  Lg = Δg + cn Sg​​, where Sg​​ is the scalar curvature of (M,g)​​ and cn​​ is an explicit numerical constant. We consider in this talk the second conformal eigenvalue of (M, [g])​​ which is defined as the infimum, over all metrics h​​ in the conformal class of [g]​​, of the renormalised second eigenvalue of Lh​​. In dimensions larger than 11 it was proven by Ammann and Humbert that the second conformal eigenvalue is attained provided (M,g)​​ is not locally conformally flat.  In this talk we investigate the lower-dimensional case 3 ≤ n ≤ 10​​. We prove that there is an open neighbourhood of the round metric on the sphere in which the second conformal eigenvalue is never attained. This is the first non-trivial non-existence result for conformal eigenvalues in dimensions larger or equal than three. This is a joint work with J. Vétois (Mc Gill).

Maria Medina - Concentrating solutions to critical competitive systems in low dimension

We will analyze the existence and the structure of different sign-changing solutions to the Yamabe equation in the whole space and we will use them to find positive solutions to critical competitive systems in dimensions 3 and 4.

Albert Clop - Sobolev flows of non-Lipschitz vector fields

In contrast to the classical Picard Theorem, for which the flow of a Lipschitz vector field is spatially bilipschitz, recent works by Jabin or also by Alberti – Crippa – Mazzucato, show that flows arising from DiPerna – Lions theory may exhibit dramatic losses of regularity and enjoy no Sobolev smoothness, even of fractional order. Between these two extreme situations, one can find many vector fields arising from Geometric Function Theory or also from Fluid Mechanics. In this talk we will review these examples and explore what can be done to understand the Sobolev smoothness of their flows.

Francesca Da Lio - Conservation Laws for Palais-Smale Relaxations of Conformally Invariant Problems

In a recent joint work with Tristan Rivière [1] we show that p-harmonic systems with antisymmetric potentials of the form

(1)         div((1+|∇ u|2)p/2 -1 ∇ u) = (1+|∇ u|2)p/2 -1Ω⋅ ∇ u​​ in 𝔻2

(where 𝔻2​​ is the unit disk in ℝ2​​ and  is antisymmetric), can be written in divergence form as a conservation law

-div((1+|∇ u|2)p/2 -1 A ∇ u) = ∇⊥ B ⋅ ∇ u​​ in 𝔻2​​ 

This extends to the p-harmonic framework the pioneering work by Rivière [3] for p = 2. We recall that (1) represents the general form of the Euler-Lagrange equation associated to the p-Energy

(2)                            Ep(u):= ∫𝔻2 (1+|∇ u|2)p/2 dx2                          

among maps taking values into a closed manifold 𝒩​↪​​ℝm​​.

Conservation laws have showed to be a central object in bubble tree analysis. In this talk we are going to explain how they can be used to tackle problems related to energy quantization, necklessness property and Morse index stability. Actually in the paper [2] we developed a new and quite robust method based on the existence of such conservation laws that allow us to show the upper-semi-continuity of the Morse index plus the nullity of critical points to conformally invariant variational problems in 2-D under weak convergence even in the case of degenerating Riemann surfaces when the length of the images of the forming collars is below some threshold.

[1] Da Lio, Francesca and Rivi`ere Tristan, Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications, arXiv: 2311.04029.

[2] Da Lio, Francesca; Gianocca Matilde and Rivi`ere Tristan, Morse Index Stability for Critical Points to Conformally invariant Lagrangians, arXiv:2212.03124.

[3] Rivière, Tristan, Conservation laws for conformally invariant variational problems, Invent. Math. 168 (1) (2007) 1-22.

Paul Laurain - Stability of the Yang-Mills Connections Index

After introducing the setting of Yang-Mills connection, I will review some classical works on Yang-Mills in critical dimension (dimension 4), such as the gauge fixing by Uhlenbeck and the energy quantization. Then, I will demonstrate how, with Mr. Gauvrit, we have proved the semi-continuity for the index of sequences of Yang-Mills connections. Finally, I will give some perspectives on the construction of non-selfdual connections.

Pierre Bousquet - Topological singularities in nonlinear Sobolev spaces

In the setting of the Sobolev space Wk,p(Bm;N)​​ from the unit ball Bm​​ into a compact manifold N​​, we present a

new approach to identify those Sobolev maps which can be approximated by a sequence in C∞​​(\overline{B}m​​;N).

This is a joint work with Augusto Ponce and Jean Van Schaftingen.

Michał Miśkiewicz - Perspectives on the p-harmonic map flow

The p-harmonic map flow is a flow of maps u : M → N between two given manifolds, and it is designed to decrease the Dirichlet p-energy (i.e. the integral of the gradient raised to the p-th power) in time. It is useful in studying p-harmonic maps, which are simply its stationary solutions. Since the pioneering work of Eells-Sampson ('64) and Chen-Struwe ('89), much is known about the special case p=2, but the general case is still full of open questions. In addition to difficulties related to the p-Laplace operator, a major obstacle seems to be the lack of a local energy monotonicity formula (due to Struwe in the case p=2). I will describe a new, alternative formulation of the p-harmonic map flow, based on the homogeneous (a.k.a. game-theoretic) p-Laplace operator. This homogeneous formulation indeed implies a Struwe-type monotonicity formula, but it comes with new fundamental challenges.  The talk is based on joint work with Erik Hupp (arXiv:2308.16096).

Melanie Rupflin - Quantitative estimates for the Dirichlet energy

In the analysis of variational problems it is often important to understand not only the behaviour of exact minimisers and critical points, but also of maps that almost minimise the energy  or that almost solve the associated Euler-Lagrange equations.

It is in particular natural to ask whether the distance of an almost minimiser to the nearest minimising state is controlled in terms of the energy defect and whether such a result not only holds in a qualitative, but in a sharp quantitative way.

In this talk we will discuss this and related questions for the classical Dirichlet energy of maps from surfaces into manifolds, in particular in the simple model problem of maps from the sphere S2​​ to itself, for which minimisers (to given degree) are simply given by meromorphic functions in stereographic coordinates.

Volker Branding - On conformal biharmonic maps and hypersurfaces

Biharmonic maps are a fourth order generalization of the well-studied harmonic map equation. They can be characterized as critical points of the bienergy for mapsbetween two Riemannian manifolds. While the energy for maps, whose critical points are precisely harmonic maps, is invariant under conformal transformations for a two-dimensional domain the bienergy is not

invariant under conformal transformations in any dimension.

In this talk we will introduce a version of the bienergy that is conformally invariant on four-dimensional manifolds and whose critical points are called conformal biharmonic maps.  We will present the basic properties of conformal biharmonic maps with particular attention to conformal biharmonic hypersurfaces in space forms and the stability of conformal biharmonic hyperspheres. Moreover, we will point out many surprising differences between biharmonic and conformal biharmonic maps.  This is joint work with Simona Nistor and Cezar Oniciuc.

Istvan Prause - Probabilistic limit shapes and harmonic envelopes

Limit shapes are deterministic surfaces in ℝ3​​ which arise in the macroscopic limit of discrete random surfaces associated to various probability models such as domino tilings, random Young tableaux or vertex models. The limit surface is a minimiser of a variational problem with a surface tension which encodes the local entropy of the model. I'll present a geometric "harmonic envelope method" which applies to a variety of models. I'll illustrate the method in detail on random Young tableaux and non-intersecting Brownian bridges. The talk is based in part on joint work with Rick Kenyon.

Christof Melcher - Transformation method for the stochastic Landau-Lifshitz-Gilbert equation

We shall discuss local well-posedness and bubbling of the stochastic Landau-Lifshitz-Gilbert equation with infinite dimensional noise. Our approach is based on random unitary transformations that yield a magnetic Landau-Lifshitz-Gilbert with rough coefficients. This is joint work with Ben Goldys and Chunxi Jiao.

Gian Paolo Leonardi - Free-boundary properties of almost-minimizers of the perimeter in non-smooth domains

Given an open n-dimensional set Ω​​ with Lipschitz boundary, a set E​​ is an almost-minimizer of the relative perimeter if it minimizes the functional P(E,Ω)​​ (roughly speaking, the (n-1)​​-area of ∂E ∩ Ω​​) among local competitors, up to a suitably quantified error. While interior regularity theory for almost-minimizers has been established since 1984, much less is known about the boundary behavior even of perimeter minimizers, when the boundary of Ω​​ is not at least of class C1,1​​. We present some results in this direction: a boundary monotonicity formula, that is valid under a so-called visibility property of Ω​​ at a given point x∈ ∂Ω​​, and a vertex-skipping property for almost-minimizers in 3-dimensional convex domains, under no extra smoothness assumptions on ∂Ω​​. The optimality of the restriction to dimension 3 of the second result will also be discussed. This research is in collaboration with Giacomo Vianello (UniTN).

Ugo Gianazza - Moduli of continuity for solutions to degenerate phase transitions

The classical two-phase Stefan problem is an archetypal free boundary problem that models a phase transition at constant temperature. It consists of solving the heat equation (or nonlinear variants of it) in the solid and liquid phases, coupled with the so-called Stefan condition at the a priori unknown interface separating them. This condition corresponds to an energy balance, prescribing the proportionality between the jump of the heat flux across the free boundary and its local velocity.

In its weak form, for which any explicit reference to the free boundary is absent, the nonlinear problem can be formulated as

(1)  ∂t β(u) - div(|Du|p-2Du) ∋ 0​​        weakly in ET,

for a function u:ET → ℝ​ representing the temperature.

Here, ET:=E×(0,T]​​, for some open set ​​​E⊂ ℝN​​​​, N∈ℕ​​​​, p>2​​, and T>0​​,

whereas β(⋅)​​​​ is the maximal monotone graph defined by

(2)                       s​​           if s>0​,

          β(s) =     [-ν,0]​​     if s=0​,

                          s-ν​​        if s<0​,

​for a positive constant ν​​, the latent heat of the phase transition, representing the aforementioned proportionality ratio. 

We substantially improve in two scenarios the current state-of-the-art modulus of continuity for weak solutions to (1)-(2): for p=N≥ 3​​, we sharpen it to

ω(r) ≈ exp(-c|ln r|1/N ),​

and for p>max{2,N}​​, we derive an unexpected Hölder modulus.

This is a joint work with Naian Liao (Universität Salzburg, Austria) and José Miguel Urbano (KAUST, Saudi Arabia and University of Coimbra, Portugal)

Vincent Millot - Fractional multiphase transitions & nonlocal minimal partitions:  closed and open questions.

I will present a convergence result for solutions of Allen-Cahn type systems with a multiple-well  potential involving the usual fractional Laplacian in the regime of the so-called nonlocal minimal surfaces.  

In the singular limit, solutions converge in a certain sense to stationary points of a nonlocal (or fractional) energy for partitions of the domain with (in general) non homogeneous surface tensions.  

Then I will present partially regularity results and open questions concerning the limiting problem underlying the new features compared to classical minimal partition problems. This talk is based on joint works with Thomas Gabard. 

Jean-Francois Babadjian - Uniqueness and characteristic flow for a non strictly convex singular variational problem

This talk addresses the question of uniqueness of the minimizers of a convex but not strictly convex integral functional with linear growth in a two-dimensional setting. The integrand - whose precise form derives directly from the theory of perfect plasticity - behaves quadratically close to the origin and grows linearly once a specific threshold is reached. We make use of spatial hyperbolic conservation laws hidden in the structure of the problem to tackle uniqueness. Our argument strongly relies on the regularity of a vector field - the Cauchy stress in the terminology of perfect plasticity - which allows us to define characteristic lines, and then to employ the method of characteristics. Using the detailed structure of the characteristic landscape, we show that this vector field is actually continuous, save for possibly two points. The different behaviors of the energy density at zero and at infinity imply an inequality constraint on the Cauchy stress. Under a barrier type convexity assumption on the set where the inequality constraint is saturated, we show that uniqueness holds for pure Dirichlet boundary data. This is a joint work with Gilles Francfort.

Bartosz Bieganowski - Normalized ground states of nonlinear Schrodinger-type equations

We propose a simple minimization method to show the existence of least energy solutions to the normalized elliptic problem on ℝN​​. The new approach based on the direct minimization of the energy functional on the linear combination of Nehari and Pohozaev constraints intersected with the closed ball in L2​​.

 Xavier Lamy  - On C1 regularity for degenerate elliptic equations in the plane

​​I will present joint work with Thibault Lacombe, where we show that Lipschitz solutions u​​ of div G(∇ u)=0​​ in a planar domain are C1​​, for strictly monotone vector fields G∈ C0(ℝ2)​​ satisfying a mild ellipticity condition. If G=∇ F​​ for a strictly convex function F​​, and 0≤ λ(ξ)≤ Λ(ξ)​​ are  the two eigenvalues of ∇2 F(ξ)​​, our assumption, stated loosely,  is that the bad set ℬ={λ=0}∩{Λ=∞}​​⊂ ℝ2​​, where ellipticity degenerates both from below and from above, is finite. This extends results by De Silva and Savin (Duke Math. J. 151, No. 3,p.487-532, 2010), which assumed either that set empty, or the larger set {λ=0​​}​  finite. Our main new input is to transfer estimates in {​​λ​​ > 0}​​ to estimates in {​​Λ <∞}​​​​  by means of a conjugate equation. When G​​ is not a gradient, the ellipticity assumption needs to be interpreted in a specific way and we provide an example highlighting the nontrivial effect of the antisymmetric part of ∇ G​​.

Evgeny Sevost'yanov - Mappings with direct and inverse Poletsky inequalities

The report is devoted to the study of mappings satisfying upper and lower inequalities on the distortion of the modulus of families of paths. Such inequalities were previously established for quasiconformal and quasiregular mappings, and for mappings of more general classes they have a more complex form. For

this reason, part of the results is devoted to obtaining minimal conditions on mappings under which modulus inequalities are valid. We have investigated the local and boundary behavior of mappings with moduli inequalities. In particular, the Holder continuity of mappings in the logarithmic sense is obtained. We have obtained a continuous boundary extension of mappings satisfying Poletsky inequalities for domains which are locally connected at the boundary, and domains with a more complex boundary structure. The case of an isolated point of the boundary of a domain is studied separately. We have proved a continuous

extension of mappings with direct and inverse moduli inequalities to an isolated point of the boundary. Some generalization of Sokhotski-Casorati-Weierstrass theorem was proved. As applications of specied estimates of the modulus distortion, new theorems on the existence of degenerate Beltrami equations are

obtained. The results mentioned above were included in the monograph

E. Sevost'yanov. Mappings with Direct and Inverse Poletsky Inequalities.

Developments in Mathematics (DEVM, volume 78). - Cham: Springer Nature

Switzerland AG, 2023, 433 p., ISBN 978-3-031-45417-2, ISBN e-book: 978-3-

031-45418-9.

Michał Kowalczyk - Generation of vortices for the Ginzburg-Landu heat flow

We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy.

We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.This is a joint work with Xavier Lamy.