Report

Teaser, summary, work performed and final results

Periodic Reporting for period 1 - SINGULARITY (Singularities and Compactness in Nonlinear PDEs)

Teaser

The emergence of singularities, such as oscillations and concentrations, is at the heart of some of the most intriguing problems in the theory of nonlinear PDEs, which lies at the heart of many questions in mathematics originating from engineering, physics, and economics. Rich...

Summary

The emergence of singularities, such as oscillations and concentrations, is at the heart of some of the most intriguing problems in the theory of nonlinear PDEs, which lies at the heart of many questions in mathematics originating from engineering, physics, and economics. Rich sources of these phenomena can be found for instance in the equations of mathematical material science. Singularities in PDEs comprise high-frequency oscillations and concentrations like jumps or fractal phenomena. These singularities correspond to the limiting behavior of the underlying model. The study of singularities (or the proof that there are none) is of paramount importance for a thorough understanding of these models.

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.

Work performed

The project has so far produced a number of results and, in fact, two of the problems outlined in the Proposal have been solved: Problems 2 has been solved by the PI, the team member Arroyo-Rabasa (postdoc) and external collaborators De Philippis (named in the proposal) and Hirsch. Problem 5 has been solved by the team member Arroyo-Rabasa. Work on the other questions is ongoing and progressing. In particular, partial results have been obtained for Problem 3 by the PI, team member Skorobogatova (PhD student), and external collaborators De Philippis and Hirsch. While some difficulties have been encountered in the work (as is to be expected for challenging questions) in general work has been progressing faster than expected and good results have been obtained.

Final results

The SINGULARITY project comprises three inter-connected research themes that deal with different aspects of the study of singularities. Each theme is based on current work by the PI and made up of individual “Problems” as milestones.

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.

Website & more info

More info: http://www.ercsingularity.org.