Giacomo Canevari : Point singularities of manifold-valued Sobolev maps and approximability by smooth maps

Manifold-valued Sobolev maps naturally arise in variational problems and models of partially ordered media, where the topology of the target can enforce the formation of singularities. These singularities may act as obstructions to familiar constructions, such as approximation by smooth maps. In this talk, given a positive integer $p$, we consider $W^{1,p}$-maps from a Euclidean domain of dimension $p+1$ into a closed Riemannian manifold $N$. The target manifold is required to satisfy suitable topological conditions; however, in contrast with previous works in this area, we do not assume that $N$ is $(p-1)$-connected. Using tools from geometric measure theory - namely, flat chains with coefficients in an appropriate homotopy group of $N$ - we associate to each map $u$ in the weak sequential closure of smooth maps an object that captures its point singularities. The vanishing of this object characterizes local strong approximability by smooth maps. This talk is based on joint work with G. Orlandi (Verona).

Kenneth Demason : Existence, Non-existence, and Regularity of Minimizers in Weighted Fractional Cluster Problems 

Recently, fractional partitioning problems have attracted significant attention: given a domain $\Omega$, how should it be divided to minimize a weighted fractional perimeter? These problems arise naturally as sharp-interface limits of fractional vectorial Allen-Cahn equations. Unlike the classical setting, where the weights correspond to certain geodesic distances between wells and satisfy a triangle inequality, in the fractional case the weights are given by the squared distance between the wells, and need not satisfy a triangle inequality.  

In this talk, we focus on minimizing clusters for the weighted fractional perimeter. Here, it is known that the energy is always lower semicontinuous, independent of the choice of surface tensions. We partially establish both existence and non-existence of minimizers in different regimes of the weights; this is in sharp contrast with the classical theory, where existence and regularity relies on a triangle inequality. When minimizers exist we prove the expected regularity: the boundary is locally smooth away from a singular set of dimension at most n-2. For almost minimizers, we obtain $C^{1,\alpha}$ regularity for some $0 < \alpha <1$. 

Federico Luigi Dipasquale : The formation of gradient-driven singular structures of codimension one and two in two-dimensions: The case study of ferronematics

We consider a variational model for ferronematics - composite materials formed by dispersing magnetic nanoparticles into a liquid crystal matrix. The model features two coupled order parameters: a Landau-de Gennes $\Q$-tensor for the liquid crystal component and a magnetisation vector field $\M$, both of them governed by a Ginzburg-Landau-type energy. The energy includes a singular coupling term favouring alignment between $\Q$ and $\M$. We report on some recent results on the asymptotic behaviour of (not necessarily minimising) critical points as a small parameter $\epsilon$ tends to zero. Our main results show that the energy concentrates along distinct singular sets: the (rescaled) energy density for the $\Q$-component concentrates, to leading order, on a finite number of singular points, while the energy density for the $\M$-component concentrates along a one-dimensional rectifiable set. Moreover, we will see that the curvature of the singular set for the $\M$-component (technically, the first variation of the associated varifold) is concentrated on a finite number of points, i.e. the singular set for the $\Q$-component. Joint work with G. Canevari (University of Verona) and B. Stroffolini (University of Naples Federico II). 

Tiziana Giorgi :  Chiral structures in chiral and achiral materials 

We study the formation of chiral structures in chiral and achiral liquid crystal systems in the simplified setting of one-dimension spacial dependence. For chiral liquid crystals, we characterize minimizers when the intrinsic pitch conflicts with the twist imposed by the boundary conditions and it is within a certain range of values. For nematic liquid crystals composed of achiral bent-core molecules, we analyze the twist-bend nematic phase. Numerical simulations based on constrained minimization explore the effects of a constant applied magnetic field. This is joint work with Carlos García-Cervera and Sookyung Joo.

Michael Goldman :  New dimensional bounds for the irrigated measure of a branched transport model coming from Ginzburg-Landau theory

In this talk, I will present results obtained in the past few years regarding the dimension of the irrigated measure for a branched transport model, describing the magnetization patterns in type-I superconductors in the regime of small external magnetic field. A compelling conjecture due to Conti, Otto and Serfaty asserts that that this dimension should be 8/5. Under the assumption of Ahlfors regularity of the measure, we recently proved together with A. Cosenza and F. Otto that the dimension cannot be larger.

Radu Ignat :  Symmetry of domain walls in Ginzburg-Landau type systems

Domain walls are topological solitons that appear in many physical systems (phase transitions, condensed matter etc). They represent transition layers between two states corresponding to the wells of a certain potential function. We will start by presenting the domain walls in the Allen-Cahn model where the potential has a finite number of wells; in particular, we will study their symmetry and energy level. Next we study the case of Ginzburg-Landau type models where the potential has a continuum of wells. For well-posedness, the divergence-free constraint is imposed on the order parameter. In this case, we will study one-dimensional symmetry of domain walls as well as the nucleation of microstructures according to the shape of the potential function.

Luana Jost :  An a priori estimate for the energy growth of minimizing entire solutions of the Ginzburg-Landau equation in $\mathbb{R}^n, n\geq 2$

In a joint work with Etienne Sandier (Université Paris-Est-Cr\'eteil), we study energy bounds of minimizers of the Ginzburg–Landau energy in balls of radius $R$ in $\R^n, n \geq 2$. The natural a-priori bound is $C R^{(n-1)}$, while the expected optimal bound is $C R^{(n-2)}\log R$. We are able to prove $C R^{ (n-4/3)}$. I will discuss the main ideas of the proof and the motivation behind the problem.

Matthias Kurzke : Local minimizers for boundary reactions

The interior reaction problem $\Delta u = f(u)$, $du/dn=0$, where $f$ is a bistable nonlinearity, is known to have no non-constant stable solutions in convex domains by a result of Casten-Holland and Matano from 1978. For the corresponding problem with a boundary reaction, $\Delta u = 0$, $du/dn=\frac1\varepsilon f(u)$, there are no non-constant stable solutions in a ball, while stable solutions exist in some dumbbell-shaped domains. We show that there exist strictly convex smooth domains where stable non-constant solutions exist for sufficiently small $\varepsilon$. The main idea is to use a constrained minimization problem to transfer information about local minimality from a finite-dimensional renormalized energy to the PDE setting. There are connections and similarities to the analysis of boundary vortices in thin magnetic films or in some classes of liquid crystals. However, in our problem the global minimizers are constant, so more work is needed to find nontrivial solutions. 

This is joint work with X. Cabre and N. Consul, https://arxiv.org/abs/2603.06435

Michał Łasica : SBV-morphisms and image similarity measures 

Perhaps the simplest way to quantitatively compare two images (of the same size) is to sum absolute differences of pixel intensities. To account for possible change of perspective or movement of objects, one may compose one of the images with a domain deformation. Recently, Ball and Horner introduced a variational image comparison model, amounting to minimization over possible deformations of a quantity composed of such integral distance and a nonlinear elasticity term. The latter ensures that deformations of finite energy are homeomorphic, the deformed image is well defined, and a minimizing deformation exists. 

In order to naturally take into account possible cuts or permutation of objects, we propose a variant of this model with a two-sided Mumford-Shah-type energy. The energy and its domain (which now includes discontinuous maps) needs to be carefully defined. On the way we revisit basic questions related to invertibility of measurable maps. Eventually, we show lower semicontinuity and existence of minimizers. Our main tool is Hajłasz's change of variables formula. 

This is joint work with Zofia Grochulska and Anastasia Molchanova.  

Benjamin Lledos :  Continuity and Uniqueness for BV minimisers 

In this talk, we study the continuity and uniqueness for the minimisers of multidimensional scalar variational problems in the space of functions of bounded variation. The integrand is assumed to be convex (but not necessarily strictly convex), with linear growth from below (but not necessarily from above), while the domain  and the boundary condition must satisfy suitable geometric and regularity conditions, such as convexity or Lipschitz continuity.

Dean Louizos :  Asymptotics for Minimizers of Landau-de Gennes with Tangential Anchoring 

In this talk I will discuss the 3D Landau-de Gennes model for nematic liquid crystal with an external magnetic field around an immersed particle. I will show that if we impose strong planar anchoring on the surface, then in the large particle limit, and assuming a weak magnetic field, we can characterize the defects that occur. We will see that only point defects can occur for minimizers in this regime.

Eve Machefert : On phase field approximation of Plateau's problem

Plateau's problem is a notorious problem in Calculus of Variations and Geometric Measure Theory. In this presentation, I will introduce a phase-field approximation of Plateau's problem, based on the coupling of the Ambrosio–Tortorelli energy with a geodesic distance penalization, which encodes the topological constraints. I will then justify this approach through a Γ-convergence result towards a formulation of Plateau's problem in codimension one, and analyze the functional by establishing existence and regularity results for minimizers. From an analytical perspective, I will also present an analysis of the limit problem and provide a characterization of quasi-minimizers in terms of John domains. Finally, this approach is implemented in a numerical framework to approximate solutions of Plateau's problem in various configurations, illustrating the efficiency and flexibility of the proposed model.

Katarzyna Mazowiecka : Existence of n-harmonic maps with prescribed degree 1 in almost round balls

We will discuss the existence of n-harmonic maps with prescribed degree on the boundary of domains of which are diffeomorphic to a ball. We will first prove that minimizers of the n-energy among maps with prescribed degree d different than 0 exist if and only if d=+-1 and the domain is a round ball. Then we will show that we can devise a min-max scheme which allows us to prove the existence of n-harmonic maps with prescribed degree equal 1 if the domain is close enough to a round ball. A bubbling analysis is necessary because of the lack of compactness of the problem. This is a joint work with Dorian Martino and Rémy Rodiac.

Benoît Merlet : A Ginzburg-Landau model with a strong penalization of the divergence and tangential anchoring

We study a two-dimensional energy functional motivated by liquid crystal models with widely varying elastic constants. The energy applies to vector fields $u_\varepsilon$ and consists of the Ginzburg-Landau functional to which we add a term penalizing the L2 energy of the divergence and with a coefficient of the order of $\log(1/\varepsilon)$. We consider configurations with tangential boundary conditions in a moderate energy regime. In this regime, the model has stronger regularity and compactness properties than those given by the classical Ginzburg-Landau theory. Indeed, we prove strong compactness of $u_\varepsilon$ and provide a detailed description of the limit : Besov regularity, traces on the boundary, vortex singularities. This work has a close connection with the study of the Aviles--Giga functional, in particular through a phenomenon of regularity by compensation. This is a joint work with Lia Bronsard, Antonín Monteil, Marc Pegon and Dominik Stantejský.

Michał Miskiewicz : Regularity of minimizing p-harmonic maps into spheres 

Regularity of minimizing p-harmonic maps, i.e.,  minimizers of the Dirichlet p-energy among maps between two given manifolds, is known to depend on the topology of the target manifold. In particular, the case of maps into spheres has been studied extensively, yet some of the most basic questions concerning maps from $\mathbb{B}^3$ into $\mathbb{S}^3$ remain open. For many years, minimizing maps in this context were known to be regular when $p=2$ or $p\geq 3$, leaving a gap in between. I will discuss how the relevant tools have developed over time (from 1984 until now) and how they eventually led to closing the gap. 

This is joint work with Andreas Gastel, Katarzyna Mazowiecka and Patryk Tokarczuk. 

Michael Novack :  A variational theory for 3D features in films and foams

We consider a geometric variational problem which models films/foams not as surfaces, but as regions with small but positive volume. In order to explain physical properties of films/foams called Plateau borders, we will present a partial regularity theorem for hypersurfaces which differs from the classical theory of Allard in two key aspects. We will discuss these differences and also, time permitting, connections to free boundary problems such as spectral minimal partitions and the two-phase Bernoulli problem. Based on joint work in preparation with F. Maggi and D. Restrepo.

Marc Pegon :  Existence, regularity, and symmetry breaking in generalized liquid drop models

In this talk, I will discuss isoperimetric problems arising from Gamow's liquid drop model for the atomic nucleus, in which the perimeter is perturbed by a non-local repulsive interaction. After reviewing classical results and conjectures related to the original model, I will present results that differ from the classical case when the repulsion kernel decays sufficiently rapidly. I will focus particularly on questions of the existence of minimizers, regularity, and symmetry-breaking phenomena.

Yannick Privat :  Optimal long-time observability of the heat equation and optimal sensor placement

It is well known that reconstructing an initial datum associated with a parabolic equation from internal measurements of its solution over a time horizon $T$, on a set $\omega$ called the observation region, is equivalent to an observability question, or more precisely to the positivity of the so-called observability constant associated with $\omega$. This constant depends on the observation region $\omega$, but also crucially on the time horizon $T$. In this talk, we are interested in the optimal placement of thermal sensors. A natural modelling choice is to look for extremal observation regions (when they exist) maximizing this observability constant. To be physically relevant, we impose a constraint on the measure of the observed region. After introducing a convex relaxation of the shape optimization problem, we determine the asymptotic behavior of maximizers as $T\to +\infty$. Using in a crucial way a quantitative bathtub principle, we prove strong convergence of the maximizers to the characteristic function of a measurable set that we characterize precisely, and we further show that this convergence is exponential. This is joint work with Idriss Mazari (Université Paris Dauphine) and Emmanuel Trélat (Sorbonne Université).

Piotr Rybka :  A few words on convergence of families of gradient flows viewed from the variational point of view 

There is no unique definition of gradient flow and we will take advantage of this fact. When we study, the question of stabilization of solutions of a perturbed gradient flow, our work would be easier if the pertubation is a gradient flow. We will show when this is possible. When we deal with a family of gradient flows, we would like to ensure that the limit is another gradient flow. We will present old and new result to this effect. We will present a few examples.

Etienne Sandier :  Minimizing entire solutions to vector Allen-Cahn equations in $\R^2$

I will report on a joint work with Peter Sternberg and recent joint work with Lia Bronsard and Peter Sternberg, where we try to describe minimizing entire solutions to the vector Allen-Cahn equation with three wells in two space dimensions. It is known that the Gamma-limit by blow-down of this problem is a minimal partition problem with weights for which minimal cones are easily described. We will see that,  depending on the weights, each minimal cone may, or may not, correspond to a minimizing entire solution.

Yana Teplitskaya :  Steiner trees and maximal distance minimizers. Recent results and open problems

I will talk about two problems: maximal distance minimizers and Steiner trees. Maximal distance minimizers concern finding a connected set of minimal length whose closed $r$-neighborhood covers a given compact set, whereas Steiner trees aim to find a minimum-length set connecting a prescribed set of points. For both problems, I briefly summarize known results and highlight the open questions and recent results. Joint work with M. Basok, D. Cherkashin, A. Gordeev, E. Paolini, N. Rastegaev and E. Stepanov.

Epifanio Virga :  A variational theory for soft shells

Three independent deformation modes can be identified for shells: these are stretching, drilling and bending. Soft shells are especially sensitive to drilling. I shall present a direct variational theory for soft shells in which each independent mode contributes a separate invariant elastic energy. Quite naturally, this turns out to be a second-grade theory, albeit in a rather indirect (perhaps also unfamiliar) way. Thus, both stress and hyper-stress tensors need to be identified and their role in interpreting the equilibrium equations as balances of forces and torques is then illuminated.

Maria Westdickenberg :  TBA

TBA

Leon Winter :  A quantitative relation between the singular set of a Sobolev section and its p-Dirichlet energy

Given a principal U(1) bundle P over a compact Riemannian surface M endowed with a connection and 1 ≤ p < 2, we give a quantitative relation between the p-Dirichlet energy of a section which is almost smooth, its singular set, and the curvature of the connection. The obtained result generalises known formulae in the flat case, where M is a smooth domain of the Euclidean space, and P the trivial bundle. In the flat case, this formula gives an answer to the lifting problem for circle valued Sobolev mappings. The new formula also implies the Gauss-Bonnet-Poincaré-Hopf theorem on compact Riemannian surfaces.

This presentation is based on joint work with Benoît Van Vaerenbergh.

Kun Lai Wu :  Entire minimizers for 2D non-abelian Ginzburg–Landau functionals

This is a joint work with Antonin Monteil. We study entire minimizers for generalized Ginzburg–Landau functionals with values in general manifolds, whose first fundamental group may be non-abelian. Nontrivial entire minimizers can be viewed as basic building blocks in the complex patterns of local minimizers away from the boundary conditions. They also help us understand higher-order asymptotic behavior of minimizers near singularities.

Kai Xiao :  Sobolev maps to manifolds: the distributional Jacobian and its applications

Starting with the observation that the strong density of smooth maps fails when considering maps that are restricted to take their values on a closed manifold, a natural question is how to characterize the maps that can be approximated with smooth maps? A good candidate to answer this in special case where the target manifold is a sphere is the distributional Jacobian, which detects the topological singularities of the map that thwart to the strong approximation with smooth maps, thus a map can be approximated if and only if its Jacobian is 0. I will introduce, when considering fractional Sobolev maps, a generalization of the Jacobian to a broader class of target manifolds that share some convenient topological properties with the sphere, and use it to characterize the closure of smooth maps in terms of restrictions on generic skeletons.

Ana Zatorska-Goldstein :  Optimization of conductivity tensor in the heat transfer problem.

We discuss the problem of finding an optimal conductivity tensor minimizing thermal compliance for a given balanced distribution of heat sources and sinks. We will discuss various a priori assumptions about the isotropy and anisotropy of the tensor. The work builds on the foundation laid by Bouchitte and Buttazzo, and the initial results are grounded in the theory of optimal transport. We propose a more direct approach. The work is done in cooperation with Piotr Rybka (MIM UW) and Tomasz Lewiński (WIL PW).