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 seminar is held by Prof. Angkana Rüland and the second lecturer is Hendrik Baers.
Find the schedule below.
Info: the seminar will take place on Fridays from 10 c.t. to 12, in room N 0.003
News: the next appointment will take place on Wednesday 18 of June at 16:00 in the room SR 2.025 (in the main building MZ). Mind the unusual place and time!
Calendar
- 11.04.2025: Hendrik Baers
Quantitative Runge approximation in the context of an inverse 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 related source-to-solution type inverse problem. One way of proving stability for such type of inverse problems is via a quantitative Runge approximation. We will investigate and prove a qualitative and a quantitative version of this approximation result. If time allows, we will then also argue how this can lead to stability estimates for the inverse problem.
- 18.04.2025: Karfreitag (i.e. no meeting)
- 25.04.2025: Camillo Tissot
Comparison Of Surface Energies In Scaling Laws
Abstract: The objective of this talk is to compare different surface energies for two-well singular perturbation problems associated with martensitic phase transformations. We show that scaling laws in the singular perturbation parameter are robust in the choice of the surface energy.
Building on Modica-Mortola type arguments we show that Lp based diffuse surface energies can always be bounded from below by a (L1 based) sharp interface penalization. We complement this by an almost matching upper bound.
- 02.05.2025: Simon Ruhland
Some classical theorems in interpolation theory
Note: The talk is based on Chapter 1 of the book "Interpolation Spaces - An Introduction" by Jöran Bergh and Jörgen Löfström.
- 09.05.2025: Hendrik Baers
Definitions and general properties of interpolation spaces
Note: The talk is based on Chapter 2 of the book "Interpolation Spaces - An Introduction" by Jöran Bergh and Jörgen Löfström.
- 16.05.2025: Manuel Heger
The real interpolation method
Note: The talk is based on Chapter 3 of the book "Interpolation Spaces - An Introduction" by Jöran Bergh and Jörgen Löfström.
- 23.05.2025: Angkana Rüland
Interpolation of Lebesgue and Sobolev spaces
Abstract: In this talk we will explore practical applications of the real interpolation method. We focus on interpolation in the context of Lebesgue and Sobolev spaces.
- 30.05.2025: Guillermo Pérez
Calderón problem for fractional Schrödinger operators on closed Riemannian manifolds
Note: The results presented in this talk are based on the homonymous paper by A. Feizmohammadi, K. Krupchyk and G. Uhlmann from the year 2024.
- 06.06.2025: (no meeting)
- 13.06.2025: Pfingstenferien (i.e. no meeting)
- 18.06.2025: Noah Piemontese-Fischer
Energy scaling laws for incompatible two-well problems
Abstract: We study the energy scaling of two-well problems for the gradient and the symmetrized gradient with surface energy in two dimensions. Typically, it is assumed that the stress-free states have a (symmetrized) rank-one connection and that the prescribed Dirichlet boundary data is a convex combination of the stress-free states. In this talk, we remove this compatibility assumption, revealing how the configuration of the stress-free states and the boundary data influences the scaling behavior. Furthermore, we discuss key arguments in the proof of the lower scaling bounds, which rely on examining the energies in Fourier space.
- 27.06.2025: Antonio Tribuzio
Local and nonlocal variational thin films
Abstract: In this talk we will see how Γ-convergence can be exploited to formally study the behaviour of thin elastic domains. Following the work of A. Braides, I.Fonseca and G. Francfort (2000), after providing compactness and integral representation results for local energies we briefly see some applications.
At the end of the talk, we will see how similar techniques can be used to study a nonlocal theory. This last part is based on a work in progress with N. Ansini.
- 04.07.2025: Daniel Linn
Entanglement principle for the fractional Laplacian
Abstract: We prove an entanglement principle for the fractional Laplacian under the assumption of super-exponential decay at infinity. The results in the talk are based on the corresponding paper of Ali Feizmohammadi and Yi-Hsuan Lin from 2024.
- 11.07.2025: Lennart Machill
Lower scaling bounds for a singular perturbation three-well problem in linearized elasticity
Abstract: In the talk, we discuss lower scaling bounds for a singular perturbation problem within the geometrically linearized theory of elasticity. We focus on a three-well problem and show that the scaling depends on both the lamination order of the prescribed Dirichlet data and the number of (non-)degenerate symmetrized rank-one directions in the symmetrized lamination convex hull. Our arguments rely on Fourier space localization methods.
The talk is based on joint work with Angkana Rüland.
- 18.07.2025: Edvin Svenungsson
Scaling laws and microstructure of a two-well model
Abstract: We study a variational problem, modelling a martensitic phase transformation, found in shape memory alloys. We begin with understanding the energy and sketching the proof of the existence of the minimizer, using the direct method and non-trivial compactness results. We finish by proving scaling laws using a self-similar construction and lower bound estimates. The results are part of my Bachelor thesis with Antonio Tribuzio and based on work by Robert V. Kohn and Stefan Müller (1994).