## Graduate Seminar on New Developments in PDE (S5B3)

In this research-oriented seminar we will discuss recent topics in Calculus of Variations and Inverse Problems.

If you are interesting in participating, contact me via email.

The second lecturer for this seminar is Lennart Machill.

**Info:** all the talks will take place on **Thursdays from 10 (c.t.) to 12** at **SR 1.007**.

**News:** next talk, on October 31, will start at **11:00**. The room is unchanged, the usual SR 1.007.

Below you find a tentative calendar of the talks.

## Calendar

- 10.10.2024:
**Vedansh Arya***(University of Jyväskylä)*

**Quantitative uniqueness to parabolic operators with applications to nodal sets**In this talk, we will discuss sharp estimate of the order of vanishing of solutions to parabolic equations with variable coefficients. For real-analytic leading coefficients, we will present localised estimate of the nodal set, at a given time-level, that generalises the celebrated one of Donnelly and Fefferman. We will also discuss Landis type results for global solutions. This is based on joint work with Agnid Banerjee and Nicola Garofalo.

Abstract:

- 17.10.2024:
**Antonio Tribuzio**

**Rigidity and flexibility in martensitic materials: the Tartar square**

**Abstract:**In recent years, the study of martensitic materials (such as certain alloys which display stunning mechanical properties like the shape-memory effect) led to the analysis of highly non-convex differential inclusions. Due to the lack of convexity, according to the prescribed regularity, there may be either many (flexibility) or one (rigidity) class of solutions.

After introducing and motivating the problem, we try to find some information about the threshold regularity between rigidity and flexibility by studying a simplified (though extrimely intresting) toy-model, the so-called Tartar square, by relaxing the problem studying scaling laws of the related singularly perturbed elastic energy.

The results presented in this talk are in collaboration with Angkana Rüland.

- 24.10.2024:
**Hendrik Baers**

**Stability of the reduction of the nonlocal Calderón problem to the local Calderón problem**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. Some of the main questions of interest are about uniqueness and stability of this reconstruction.

Abstract:

In this talk, we will consider a strongly related problem, the Calderón problem with source-to-solution data on a closed manifold, and its fractional (or nonlocal) formulation. It is known that the data in the nonlocal problem and in the local problem are strongly related. More precisely, the nonlocal data uniquely determine the local data. As a consequence, any uniqueness result in the local setting can be transferred to the nonlocal problem. We seek to quantify the uniqueness result of the reduction process, allowing then to also transfer stability results for the local setting to the nonlocal one.

- 31.10.2024:
**Pascal Steinke**

**Homogenization of Interaction Energy of Dislocations**

**Abstract:**Dislocations are a type of material defects in the crystallographic structure of metals. They form due to external forces and play a fundamental role in determining the elasticity and brittleness properties of the metal.

We consider the total interaction energy of dislocation loops placed on a homogenization-lattice, then we let the lattice spacing tend to zero. Under a well-separatedness condition, we will deduce that the asymptotic behaviour is similar to that of dipoles on the same lattice. Inspired by a previous work of R. James and S. Müller, using the notion of H-Measures introduced by L. Tartar, we will derive a limiting representation of the total interaction energy and establish its Gamma-convergence to -infinity.

Joint work with Stefan Müller.

**Note:**we will start (unusually) at**11:00**, still at SR 1.007

- 07.11.2024:
**Camillo Tissot**

**Scaling law for a discrete Tartar square**Motivated by microstructures in the modelling of shape memory alloys and their associated differential inclusions, we study a discrete Tartar square.

Abstract:

A striking property of the Tartar square is the dichotomy between rigidity of exact solutions and flexibility of approximate solutions for the associated differential inclusion.

Instead of the standard singularly perturbed model, consisting of an elastic energy plus a surface energy multiplied by a small parameter ε (which stands for a length scale), we look for minimizers of an elastic energy satisfying a discreteness condition.

In this way, by assuming that the functions involved are only defined on a grid, we 'penalize' high oscillations.

After relating this constraint to the well-studied singularly perturbed model, we analyse the scaling for a discrete Tartar square in the grid size h. This will coincide with the scaling law in ε of the singularly perturbed model.

This is based on joint work with Angkana Rüland, Antonio Tribuzio and Christian Zillinger.

- 14.11.2024:
**Lennart Machill**

**An Introduction to Nonlinear (Thermo-)Viscoelasticity at Large Strains**

**Abstract:**We discuss Kelvin-Voigt models for viscoelastic materials. To prove the existence of weak solutions, we use a variational time-discretization scheme [Mielke, Roubícek '20]. In this context, we review a result by [Healey, Krömer '09] and discuss an approximation scheme which satisfies time-discrete frame indifference. Afterwards, we include an additional coupling with a nonlinear heat equation and present a technique to show that the temperature remains positive along the evolution.

The talk is based on joint work with Rufat Badal, Manuel Friedrich, and Martin Kružík.

- 21.11.2024:
**Guillermo Pérez**

**Branched microstructures: Scaling and asymptotic self-similarity**

**Note:**the results presented in this talk come from the homonymous paper from S. Conti (2000)

- 28.11.2024:
**Thomas Häßel**

**A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity**

**Note:**the results presented in this talk come from the homonymous paper from G. Frieseke, R. D. James and S. Müller (2002)

- 05.12.2024:
**Linus Engelfried**

**tba**

**Abstract:**tba

- 12.12.2024:
**Oskar Engelfried**

**tba**tba

Abstract:

- 19.12.2024:
**tba**

**tba**

**Abstract:**tba

- 09.01.2025:
**Yi-Hsuan Lin**(*University of Duisburg-Essen*)

**tba**

**Abstract:**tba

- 16.01.2025:
**Noah Piemontese-Fischer**

**tba**

**Abstract:**tba

- 23.01.2025:
**tba**

**tba**

**Abstract:**tba

- 30.01.2025:
**tba**

**tba**

**Abstract:**tba