Arbeitsgruppe Analysis und Partielle Differentialgleichungen

Graduate seminar on Advanced topics in PDE

Organizers

Schedule

This seminar takes place regularly on Fridays, at 14:15. The seminar will take place in person in SemR 0.011. Please join the pdg-l mailing list for further information.

Oct 17 - Hjørdis Schlüter (University of Helsinki)

Title: Boundary determination in anisotropic elasticity

Abstract:

This is joint work with Joonas Ilmavirta, Mikko Salo and Daniel Windisch. I will talk about the elastic wave equation that models waves that propagate through elastic media. We are particularly interested in materials that are highly anisotropic and aim at recovering the density of mass and the so-called stiffness tensor, that capture the elasticity of the material, from exterior measurements. The exterior measurement that we consider is the Dirichlet-to-Neumann map. I will talk about a recent result, where we for generic stiffness tensor fields are able to determine the normalized stiffness tensor (where we are not able to distinguish between density and stiffness tensor) at the boundary of the medium. For this result we need a construction of special solutions and employ tools from algebraic geometry.

Oct 17, 15:45 - Organizational meeting

Oct 24 - Hendrik Baers (University of Bonn) - Joint seminar with Prof. Rüland's research group

Title: Quantitative reduction of a nonlocal Calderón type 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 the fractional (or nonlocal) and the local formulation of a Calderón type problem and discuss how these two are related. In particular, we will show that stability estimates for the local problem can be transferred to stability estimates for the nonlocal problem. The results presented in this talk are based on joint work with Angkana Rüland.

Oct 31 - Yi Zhang (Chinese Academy of Sciences)

Title: Weak solutions to Serrin's overdetermined problem

Abstract:

Serrin's classical overdetermined theorem states that if a bounded C^2-domain supports a function with a constant Laplacian that satisfies both constant Dirichlet and Neumann boundary conditions, then the domain must be a ball. In this presentation, we explore this problem further, demonstrating that the theorem's conclusion extends to a broader class of domains, including Lipschitz domains. Our method is based on theory from geometric measure theory and harmonic analysis.