Analysis group : day of seminar talks

by Carlos Alberto dos Santos (UnB, Brasili), Edward Bryden (Antwerp), Hidde Schönberger (UCLouvain), Jakob Fuchs (Dortmund)

Thursday 9 Oct 2025, 10:00 → 17:00 Europe/Brussels

E/1st floor-E.161 - Meeting room (E.161) (Marc de Hemptinne (chemin du Cyclotron, 2, Louvain-la-Neuve))

10h- 11h : Hidde Schönberger : Local boundary conditions in nonlocal hyperelasticity via heterogeneous horizons

In this talk, we consider a class of variational problems with integral functionals involving nonlocal gradients. These models have been recently proposed as refinements of classical hyperelasticity, aiming for an effective framework to capture also discontinuous and singular material effects. Specific to our set-up is a space-dependent interaction range that vanishes at the boundary of the reference domain. This ensures that the nonlocal operator depends only on values within the domain and localizes to the classical gradient at the boundary, which allows for a seamless integration of nonlocal modeling with local boundary values. Our main contribution is a comprehensive theory for the associated Sobolev spaces, including the rigorous treatment of a trace operator and Poincaré inequalities. As an application, we establish the existence of minimizers for functionals with quasiconvex or polyconvex integrands depending on heterogeneous nonlocal gradients, subject to local Dirichlet or Neumann boundary conditions. This is based on a joint work with Carolin Kreisbeck (KU Eichstätt-Ingolstadt).

11h15-12h15 : Edward Bryden : Stability for a class of three-tori with small negative scalar curvature

We define a flexible class of Riemannian metrics on the three-torus. Then, using Stern's inequality relating scalar curvature to harmonic one-forms, we show that any sequence of metrics in this family whose negative part of the scalar curvature tends to zero in L² norm has a subsequence which converges to some flat metric on the three-torus in the sense of Dong-Song. 

14h30-15h30 : Carlos Alberto dos Santos : A continuation theorem applied to problems with singular regions 

In this talk, we will present an abstract result that will be applied to prove the existence of a connected set of solutions for problems that do not necessarily have a priori boundedness of solutions and may present regions of singularities. We apply this new result to establish the existence of connected branches of strongly positive classical solutions for Dirichlet problems headed by quasilinear Schrödinger- and Carrier-type operators. A fine qualitative study of these connected branches is also presented. 

16h-17h : Jakob Fuchs : Sharp Interface Reduction of a Mesoscale Model for Two-Species Surfacatant Films

We propose a variational model for two-phase surfactant films separating aqueous and oily fluids. Considering two species of surfactant molecules we describe a phase seperation within the film. The analysis builds on a model for single species lipid biomembranes proposed by Peletier and Röger [ARMA 2009]. We prove a Gamma-convergence result in the limit of vanishing surfactant length and show that the limit inherits a phase separation and a bending energy.