Schedule

Felix Schulze

Generic regularity of area minimising hypersurfaces up to dimension 11

We prove that area-minimizing hypersurfaces are generically smooth in ambient dimension 11 in the context of the Plateau problem and of area minimization in integral homology. For higher ambient dimensions, n+1 ≥ 12, we prove in the same two contexts that area-minimizing hypersurfaces have at most an (n − 10 − εₙ)-dimensional singular set after an arbitrarily C∞-small perturbation of the Plateau boundary or the ambient Riemannian metric, respectively. This is joint work with O. Chodosh, C. Mantoulidis and Z. Wang.

Damian Dąbrowski

Favard length and quantitative rectifiability

Favard length of a planar set is the average length of its orthogonal projections. The Besicovitch projection theorem, which is one of the cornerstones of geometric measure theory, states the following: if a set E of finite length has positive Favard length, then there exists a rectifiable curve intersecting E in a set of positive length. In this talk I will discuss my recent quantification of this classical result, and its application to Vitushkin’s conjecture.

Dorian Martino

Some global properties of umbilic points of Willmore immersions in the 3-sphere

In the study of Willmore surfaces, the conformal Gauss map approach introduced by Bryant in 1984 is the key tool for the classification of (branched) Willmore spheres. However, its geometric behaviour is still not well understood in higher genus. In this talk, I will focus on the relations between the umbilic set of Willmore surfaces and the singularities of its conformal Gauss map. I will present a Gauss-Bonnet formula for the conformal Gauss map which provides a unified formulation of the known formulas giving the value of the Willmore energy of conformal transformations of minimal surfaces in the three model spaces. Joint work with Nicolas Marque.

Jonas Hirsch

Isometric immersions and weak solutions to the Darboux equation

Joint work with Wentao Cao, and Dominik Inauen

I would like to present some recent observations on studying the Darboux equation, a fundamental PDE arising in the theory of isometric immersions of two-dimensional Riemannian manifolds into ℝ³, in the low-regularity regime. We introduce a notion of weak solution for u ∈ C¹,θ with θ > 1/2, and show that the classical correspondence between solutions of the Darboux equation and isometric immersions remains valid in this regime. The key ingredient is an extension of the classical flatness criterion to Hölder continuous metrics, achieved via an analysis of a weak notion of Gaussian curvature.

Karol Bołbotowski

Zolotarev metric as an alternative for Wasserstein

In the late ‘70s V.M. Zolotarev proposed a family of distances between two probabilities by naturally extending the Wasserstein-1 distance to higher orders: he proposed to bound the Lipschitz constant of the derivatives of the potential. However, the optimal transport (OT) reformulation of the Zolotarev distance has not been available beyond the first order. In my talk I will demonstrate how a PDE motivation revolving around optimal elastic structures has led us to a new (OT) framework for the second-order Zolotarev distance. At the core of the link will lie a probability distribution that optimally dominates the data for the convex order. The new (OT) formulation opens new possibilities of applying Zolotarev-2 metric, previously exclusive to Wasserstein. It also paves the way to equivalence between the Zolotarev-2 and Wasserstein-2 distances in the form of sharp inequalities.

This is a joint work with Guy Bouchitté (Université de Toulon).

Michał Miśkiewicz

Homotopy type of maps with an upper bound on the rank of derivative

Given a C¹-map F: Sⁿ⁺¹ → Sⁿ, does the upper bound rank DF ≤ k force F to be homotopically trivial? Or more precisely, what is the largest k for which the answer is positive? The topology here is well-understood, as the homotopy type of F is determined by classical algebraic invariants. However, the effects of analytical degeneracy are hard to study, especially for n > 3, when one is forced to work with the Z₂-valued Steenrod--Hopf invariant. I will review some motivation coming from Gromov's work on Carnot-Carathéodory spaces, the main new ideas due to Guth, and finally our own recent contribution.

Joint work with  E. Eyeson, P. Goldstein and P. Hajłasz.

Arghir Dani Zarnescu

A variational approach to fluids

We consider a construction proposed by A. Acharya  in QAM 2023, LXXXI(1) that builds on the notion of weak solutions for incompressible fluids to provide a scheme that  generates variationally a certain type of dual solutions. If these dual solutions are regular enough one can use them  to recover standard solutions. The scheme provides a generalisation of a construction of Y. Brenier for Euler. We rigorously analyze the scheme, extending the work of Brenier for Euler, and also providing an extension of it to the case of Navier-Stokes equations.

This is joint work with A. Acharya and B. Stroffolini.

Anastasia Molchanova

Continuity of Capacity and Applications to Electroelasticity

We study the continuity of variational capacity under weak convergence of Sobolev mappings, proving both lower and upper semicontinuity results under suitable regularity assumptions on the underlying domains. As an application, we analyze the equilibrium problem for charged hyperelastic solids, where mechanical deformation interacts with electrostatics through a capacitary term in the energy functional, and we establish the existence of minimizers for the coupled energy.

Hoai Minh Nguyen

Characterizations of the Sobolev norms and the total variation via nonlocal functionals, and related problems

We  briefly discuss the contribution of Haiim Brezis and his co-authors on the characterizations of the Sobolev norms and the total variation using non-local functionals. Some ideas of the analysis are given and new results are presented. 

Armin Schikorra

On s-Stability of W^{s,n/s}-minimizing maps between spheres in homotopy classes

We consider maps between spheres S^n to S^\ell that minimize the Sobolev-space energy W^{s,n/s} for some s \in (0,1) in a given  homotopy class. The basic question is: in which homotopy class does a minimizer exist?  This is a nontrivial question since the energy under consideration is

conformally invariant and bubbles can form. Sacks-Uhlenbeck theory tells us that minimizers exist in a set of  homotopy classes that generates the whole homotopy group \pi_{n}(\S^\ell). In some situations explicit examples are known if  n/s = 2 or s=1.

In our talk we are interested in the stability of the above question in dependence of s. We can show that as s varies locally, the set of  homotopy classes in which minimizers exist can be chosen stable. We  also discuss that the minimum W^{s,n/s}-energy in homotopy classes is continuously depending on s.

Joint work with K. Mazowiecka (U Warsaw)

Cyrill Muratov

Skyrmions in ultrathin magnetic films: an overview

I will present an overview of the current results on existence and asymptotic properties of magnetic skyrmions defined as topologically nontrivial maps of degree +1 from the plane to a sphere which minimize a micromagnetic energy containing the exchange, perpendicular magnetic anisotropy  and interfacial Dzyaloshinskii-Moriya interaction (DMI) terms. In ultrathin films, the stray field energy simply renormalizes the anisotropy constant at leading order, but in finite samples it also produces additional non-trivial contributions at the sample edges, promoting nontrivial spin textures. Starting with the whole space problem, I will first discuss the existence of single skyrmions as global energy minimizers at sufficiently small DMI strength. Then, using the quantitative rigidity of the harmonic maps I will present the asymptotic characterization of single skyrmion profiles both in infinite and finite samples. Lastly, I will touch upon the question of existence of multi-skyrmion solutions as minimizers with higher topological degree and present recent existence results obtained jointly with T. Simon and V. Slastikov.

Luca Martinazzi

Variational Problems under topological constraints

The seminal work of Brezis-Coron (1983) for 2-dimensional harmonic maps introduces an estimate that leads to the existence of harmonic maps minimizing in different homotopy classes. This had important consequences dimension 3, leading to the celebrated proof by Tristan Rivière of the existence of everywhere discontinuous harmonic maps and the partial regularity result of Hardt-Lin-Poon for minimisers of the axially symmetric relaxed Dirichlet energy.

We will discuss what analogies and differences arise when following a similar path for the 1-dimensional half-harmonic map case and for the 4-dimensional Yang-Mills functional. The talk will be based on joint works with Ali Hyder (TIFR Bangalore) and Tristan Rivière (ETH Zurich).

Antonin Monteil

Singular harmonic maps on planar domains into compact manifolds and their tangent cones 

We will begin by reviewing some results on harmonic maps u into Riemannian manifolds and their singularities. In dimension 2, harmonic maps are known to be smooth, which leads to a topological obstruction: a non-contractible boundary condition in the target manifold 𝒩 cannot be extended from the boundary to the interior of a simply connected domain.

To address this difficulty, one can either minimize the energy on a perforated domain, or relax the constraint u ∈ 𝒩 ⊂ ℝᵏ by introducing a potential W vanishing on 𝒩, as in the Ginzburg–Landau model of Bethuel–Brezis–Hélein when 𝒩 = 𝕊¹. This yields maps that are harmonic away from a finite set of point singularities, where the energy degenerates.

We will discuss the topological and geometrical features of these singularities, and address the question of uniqueness of blow-down limits near singular points. The results presented are joint work with Rémy Rodiac, and with Jean Van Schaftingen.

Adriano Pisante

Torus vs split solutions for the Landau de Gennes model.

We report on some recent works (in collaboration with F. Dipasquale and V.Millot) about the study of global minimizers of a continuum Landau-De Gennes energy functional for nematic liquid crystals in three-dimensional domains. First, we discuss absence of singularities for minimizing configurations under norm constraint, as well as absence of the isotropic phase for the unconstrained minimizers, together with the topological character of the related biaxial escape phenomenon. Then, we discuss the previous properties under both the norm and the axial symmetry constraints, showing that in this case only partial regularity is available, away from a finite set located on the symmetry axis where we describe precisely the asymptotic profile around singular points. Moreover, we discuss presence/absence/coexistence of smooth and singular minimisers  in a fixed symmetric domain as the boundary data vary. Finally, the same properties are investigated under radial anchoring when the domain is suitably deformed. 

Riikka Korte

Conformal transformations of metric measure spaces

I will discuss the recent results on conformal deformations of metric measure spaces. Inspired by the stereographic projection and its inverse, the deformations that transform unbounded spaces into bounded ones are called sphericalizations, and transformations that transform bounded spaces into unbounded ones are called flattenings. It is possible to construct sphericalizations and flattenings that preserve for example uniformity of a domain, doubling property, Poincaré inequality, p-energy and/or Besov energy. Thus one can transform for example a Dirichlet boundary value problem on an unbounded domain into an equivalent domain in a bounded domain. I will discuss our recent results that are based on joint work with A. Björn, J. Björn, R. Gibara, S. Rogovin, N. Shanmugalingam and T. Takala. 

Andrea Pinamonti

Some regularity results for balance laws and applications to the Heisenberg group

In this talk we prove Hölder regularity of any continuous solution 𝑢 to a 1d scalar balance law, when the source term is bounded and the flux is nonlinear of order 𝑝 ∈ℕ with 𝑝 ⩾2. Finally, we apply a refinement of the previous result to provide a new proof of the Rademacher theorem for intrinsic Lipschitz functions in the first Heisenberg group. The talk is based on a joint paper with L. Caravenna and E. Marconi.

Zofia Grochulska

Attempt at interpolating Sobolev spaces on arbitrary planar domains

A normed vector space A is an interpolation space between A₀ and A₁ if, roughly, it is contained in between these spaces and a certain additional desirable property holds. Namely, continuity of a linear operator on A₁ and on A₀ should imply its continuity on A. I will focus on the real interpolation method (introduced by Peetre) and apply it to Sobolev spaces on ℝⁿ (Adams–Fournier, DeVore–Scherer). Next, I will discuss both known and new results on interpolating Sobolev spaces on planar domains.

This is work in progress with Pekka Koskela and Riddhi Mishra (both from University of Jyväskylä).

Daniel Faraco

Existence of Minimizers in Elasticity Beyond Polyconvexity in 2D

Hyperelasticity is the branch of continuum mechanics in which the Piola–Kirchhoff stress tensor can be expressed in terms of a strain energy density. Equilibrium configurations are then naturally linked to minimizers of vectorial variational problems. Classically, existence of minimizers depends on convexity properties of the stored energy. In the vectorial case, Morrey showed in the 1950s that quasiconvexity is the essential condition.

However, frame indifference and the requirement of infinite energy cost for vanishing volumes rule out convexity of the strain energy. In the late 1970s, J.M. Ball introduced the notion of polyconvexity, lying strictly between convexity and quasiconvexity, and showed that it is consistent with the physical principles of hyperelasticity. Polyconvexity is also mathematically tractable, and in the past four decades it has served as the main structural condition for existence results in nonlinear elasticity, inspiring a substantial literature in both applied and pure mathematics.

In this talk I will describe recent progress showing that existence of minimizers can be established beyond polyconvexity in 2D, by drawing on ideas from geometric function theory

This is a joint work with K.Astala (Helsink University) A.Guerra (Univesity of Cambridge), A.Koski (Aalto University, Helsinki) and J.Kristensen (Oxford).