Winter Semester 2023/2024

Below, you find the list of seminar talks we had in the winter semester 2023/2024.

Calendar

• 13.10.2023: Antonio Tribuzio
  
  Energy barriers for boundary nucleation in a two-well model
  
  Abstract: Shape-memory alloys are specific materials that, e.g. during a cooling process, change their crystalline structure. For this, their internal elastic energy is a multi-well functional acting on the deformation gradient. This trasformation is often initiated by the formation of a small inclusion of deformed material (nucleation).
  In this talk, we study scaling laws for a double-well singularly-perturbed elastic energy in which the inclusion of deformed material is constraint in the halfspace and has prescribed volume.
  This problem is a variant of the isoperimetric problem with an additional (nonlocal and anisotropic) bulk term given by the elastic energy. We will see how the relation between the anisotropy of the material and the constraint affects the scaling.
  This is a joint work (in preparation) with Konstantinos Zemas.
   
• 20.10.2023: Hendrik Baers
  
  Instability of the Fractional Calderón Problem
  
  Abstract: The Calderón Problem is one of the classic examples of an inverse problem. It is about determining the conductivity of a medium by making voltage and current measurements on its boundary. We consider the fractional formulation of the problem and prove exponential instability.
  The first part of the talk focuses on the interplay of regularity of solutions and instability. We see (on a very formal level) that high regularity of solutions imply strong compression of the forward problem, which in turn leads to instability for the inverse problem.
  In the second part of the talk we prove the regularity result.
   
• 27.10.2023: Noah Piemontese-Fischer
  
  Infinite-dimensional inverse problems with finite measurements
  
  Abstract: We consider a general inverse problem, where the forward map is between two infinite-dimensional Banach spaces. Often these problems are inherently ill-posed, as small errors in the data amplify exponentially in the reconstruction. Nonetheless, Lipschitz stability can be obtained if it is assumed that the unknown of the inverse problem lies in a finite-dimensional subspace. From a practical point of view, this result is still unsatisfactory because the infinite-dimensional data can only be approximated through finite measurements.

  This talk is an exploration of the homonymous article by G. S. Alberti and M. Santacesaria (2022). Their work provides a general framework to study uniqueness, stability and reconstruction of inverse problems when we only have access to a finite-dimensional approximation of the data. After introducing the main result of the article, the framework is applied to the Calderón Problem. Furthermore, a reconstruction algorithm is presented which is rooted in the Landweber iteration.

• 03.11.2023: Camillo Tissot
  
  Lower scaling bounds for a class of two-well problems
  
  Abstract: We study the scaling behavior of a class of two-well problems for higher order, homogeneous linear differential operators. To this end, we first deduce general lower scaling bounds which are determined by the vanishing order of the symbol of the operator on the unit sphere in direction of the associated element in the wave cone. This result illustrates that for the two-well problem for higher order operators new scaling laws emerge which are determined by the Fourier symbol in the direction of the wave cone.
   
• 13.11.2023: Lennart Machill (Uni Münster)
  
  Derivation of von Kármán theories for viscoelastic solids
  
  Abstract: In the talk, we derive an effective one-dimensional limit from a three-dimensional Kelvin-Voigt model for viscoelastic thin-walled beams, in which the elastic and the viscous stress tensor comply with a frame-indifference principle. The model satisfies a variational structure, and we investigate the model by means of gradient flows in metric spaces. In particular, the dimension reduction is performed by using evolutionary Γ-convergence. We also address the scaling of the Korn constant for a specific nonlinear Korn inequality on thin domains. The talk is based on joint work with Rufat Badal and Manuel Friedrich.
  
  Note: this talk will take place in SR 0.003
   
• 17.11.2023: Matteo Ravot-Licheri
  
  A new type of CGO solutions and its applications in corner scattering
  
  Note¹: the results presented in this talk come from the homonymous paper from Jingni Xiao (2022)
  Note²: this talk will take place in room 2.040
   
• 24.11.2023: Wenhui Shi (RWTH Aachen)
  
  Second order expansion for the nonlocal perimeter functionals
  
  Abstract: In this talk we consider a family of energies which are motivated from modeling  pattern formation for ferromagnetic thin films with perpendicular anisotropy, and also related to the second order expansion for the nonlocal perimeter functional. We will discuss the $\Gamma$-convergence of the energies and the minimization problem for the limiting energy. The analysis is based on an exploitation of the so-called autocorrelation function. This is based on joint work with Hans Knüpfer.
   
• 01.12.2023: Giovanni Covi
  
  Geometrical optics solutions for nonlocal scattering
  
  Abstract: We will discuss the inverse problem for nonlocal scattering, and relate it to the fractional Helmholtz equation. For this problem, we will show how to construct approximate geometrical optics solutions and how to upgrade them to exact solutions. We believe that this technique, which is close in spirit to the one used for the analogous local problem, will provide a useful alternative to the established Runge approximation technique for fractional inverse problems. If time allows, we will show how to generalize the construction to the case of  isotropic fractional elasticity. This is an ongoing joint work with professors Maarten de Hoop and Mikko Salo.
   
• 08.12.2023: Xiapeng Cheng
  
  Singular solutions of elliptic equations and the determination of conductivity by boundary measurements
  
  Abstract: In this talk, we present results from a paper by Giovanni Alessandrini (1990). For the recovery of the conductivity from Dirichlet-to-Neumann map, we will present stability results for the determination of conductivity and its derivatives on the boundary. The method relies on construction of solutions of the conductivity equation with one isolated singularity with prescribed asymptotic behavior.
   
• 19.01.2024: Vanessa Hüsken (Uni Duisburg-Essen)
  
  A geometrically nonlinear Cosserat model for micro-polar elastic solids
  
  Abstract: Since the beginning of the 1900s, linearized Cosserat elasticity is well known and often used in the engineering community for modeling micro-polar elastic solids. But from a mathematician’s perspective a geometrically nonlinear version of such models is interesting. Existence of solutions for the latter has been known for about 20 years, while regularity questions were first investigated only during the last four years.
  In the first half of the talk, we will get an introduction to the model itself as well as an overview over recently developed different regularity results for Cosserat energy minimizers and critical points (i.e. weak solutions of the Euler-Lagrange equations). We will see, how classical regularity theory for harmonic maps into manifolds is an essential tool in deriving those results.
  At the same time, the geometric nature of the model’s nonlinearity allows not only regular (Hölder continuous) but also quite singular solutions to exist, which we will focus on in the second half of the talk.
  
  Note: you can find a PDF version of the abstract here
   
• 26.01.2024: Alice Marveggio
  
  Stability of planar multiphase mean curvature flow beyond a circular topology change
  
  Abstract: The evolution of a network of interfaces by mean curvature flow features the occurrence of topology changes and geometric singularities. As a consequence, classical solution concepts for mean curvature flow are in general limited to a finite time horizon. At the same time, the evolution beyond topology changes can be described only in the framework of weak solution concepts (e.g., Brakke solutions), whose uniqueness may fail.
  Following the relative energy approach, we prove a quantitative stability estimate holding up to the singular time at which a circular closed curve shrinks to a point. This implies a weak-strong uniqueness principle for weak BV solutions to planar multiphase mean curvature flow beyond circular topology changes. We expect our method to have further applications to other types of shrinkers.
  
  This talk is based on a joint work with Julian Fischer, Sebastian Hensel and Maximilian Moser.