Graduate Seminar on New Developments in PDE (S5B3)
In this research-oriented seminar we discussed recent topics in Calculus of Variations and Inverse Problems.
The second lecturer for this seminar is Lennart Machill.
Calendar
- 10.10.2024: Vedansh Arya (University of Jyväskylä)
Quantitative uniqueness to parabolic operators with applications to nodal sets
Abstract: 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.
- 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
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. Some of the main questions of interest are about uniqueness and stability of this reconstruction.
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
Abstract: Motivated by microstructures in the modelling of shape memory alloys and their associated differential inclusions, we study a discrete Tartar square.
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
Note: The talk is based on the paper "A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity" by G. Friesecke, R. D. James, and S. Müller from the year 2002. The focus of the talk is the proof of the rigidity estimate.
- 05.12.2024: Linus Engelfried
Comparison between laminates and branching in the Tatar square
Note: The results discussed in the talk are part of the bachelor's thesis and are based on the articles: "On the energy scaling behaviour of singular perturbation models with prescribed dirichlet data involving higher order laminates" and "On Scaling Laws for Multi-Well Nucleation Problems Without Gauge Invariances" both from Prof. Rüland and Dr. Tribuzio.
- 12.12.2024: Oskar Engelfried
Comparison between laminates and branching in L1-based elastic energies
Note: The results discussed in the talk are part of the bachelor's thesis and are based on the articles "An optimal scaling law for finite element approximations of a variational problem with non-trivial microstructure" from Andrew Lorent and "On the energy scaling behaviour of singular perturbation models with predescribed dirichlet data involving higher order laminates" from Rüland and Tribuzio.
- 19.12.2024: Christmas greetings (i.e. no meeting)
- 09.01.2025: Yi-Hsuan Lin (University of Duisburg-Essen)
Entanglement principle for the fractional Laplacian with applications to inverse problems
Abstract: We prove an entanglement principle for fractional Laplace operators on Rn for n≥2 as follows; if different fractional powers of the Laplace operator acting on several distinct functions on Rn, which vanish on some nonempty open set O, are known to be linearly dependent on O, then all the functions must be globally zero. This principle is known to be true on compact Riemannian manifolds without boundary. Our main result extends the principle to the noncompact Euclidean space under suitable decay conditions at infinity. We also present applications of this principle to solve new inverse problems for recovering anisotropic principal terms as well as zeroth order coefficients in fractional polyharmonic equations. Our proof of the entanglement principle uses the heat semigroup formulation of fractional Laplacian to establish connections between the principle and the study of several distinct topics in analysis, including interpolation properties for holomorphic functions under certain growth conditions at infinity, meromorphic extensions of holomorphic functions on a subdomain, as well as support theorems for spherical mean transforms on Rn that are defined as averages of functions over spheres. This is a joint work with Ali Feizmohammadi.
- 16.01.2025: Noah Piemontese-Fischer
Relaxation and scaling of a singularly perturbed, incompatible two-well energy
Abstract: In this talk, we investigate the scaling behavior of a singularly perturbed two-well energy (for the gradient) with incompatible stress-free states. This problem is motivated by the study of martensitic transformations in alloys and builds on the seminal work of Kohn and Müller. We discuss that despite the incompatibility, the energy exhibits the same ϵ2/3 scaling law as observed in the compatible setting. I will provide intuition behind this scaling behavior and outline additional challenges posed by incompatibility.
This talk is based on my master’s thesis, supervised by Angkana Rüland and Antonio Tribuzio.
- 23.01.2025: Yavar Kian (LMRS, Université de Rouen Normandie)
Determination of quasilinear terms from restricted data
Abstract: In this talk we will consider the inverse problem of determining a nonlinear term appearing in some class of nonlinear equations. This class of problems have received a lot of attention among the mathematical community due to both the significance of their applications and their high nonlinearity. Most of the results in that category have been stated with data corresponding to the knowledge of the corresponding Dirichlet-to-Neumann map which requires to much data in is difficult to compute numerically. The main goal of this talk is to consider this class of problems with some important restriction on the data used so far. Our main objective is to consider data that will be easier for numerical computation in order to establish theoretical results that can be jointly considered with numerical reconstruction.
- 30.01.2025: Greetings (i.e. no meeting)