Periodic Reporting for period 3 - SINGULARITY (Singularities and Compactness in Nonlinear PDEs)
Okres sprawozdawczy: 2021-04-01 do 2022-09-30
For instance, in certain materials (e.g. CuAlNi crystals) one can observe microstructure, i.e. finely layered material phases. These materials have many important applications, for example as shape-memory alloys, where a previous deformation is “remembered” by a specimen in the form of such fine oscillations between material phases. The specimen will then return to its original shape once it is heated above a certain temperature. Another example of singularity formation is shear localization, e.g. shear banding, in (perfect) elasto-plasticity theory. The shape of these concentration effects has strong implications for the macroscopic be- havior of the material and its engineering properties. We also mention shocks in multi-dimensional systems of conservation laws, and the more tangentially related turbulent fluid flow.
The SINGULARITY project will investigate singularities through innovative strategies and tools that combine the areas of geometric measure theory with harmonic analysis. The potential of this approach is far-reaching and has already led to the resolution of several long-standing conjectures as well as opened up new avenues to understand the fine structure of singularities.
Theme I investigates condensated singularities, i.e. singular parts of (vector) measures solving a PDE. A powerful structure theorem was recently established by the PI and De Philippis, which will be developed into a fine structure theory for PDE-constrained measures.
Theme II is concerned with the development of a compensated compactness theory for sequences of solutions to a PDE, which is capable of dealing with concentrations. The central aim is to study in detail the (non-)compactness properties of such sequences in the presence of asymptotic singularities, for instance in relation to the Bouchitt ́e Conjecture in shape optimization.
Theme III investigates higher-order microstructure, i.e. nested periodic oscillations in sequences, such as laminates. The main objective is to understand the effective properties of such microstructures and to make progress on pressing open problems in homogenization theory.