Periodic Reporting for period 4 - SINGULARITY (Singularities and Compactness in Nonlinear PDEs)
Período documentado: 2022-10-01 hasta 2024-03-31
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.
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.
Over its running time of six years, the SINGULARITY project has achieved a large fraction of its envisaged goals, sometimes going far beyond what was originally envisaged. Nearly 50 scientific papers have been published and almost 100 scientific talks have been delivered by the team members at workshops, seminars, and international conferences.
The most important results of the project are the following:
(1) Strong rectifiability results for PDE-constrained measures, providing a unified and generalized approach to a number of central rectifiability questions in geometric measure theory (e.g. for BV, BD, varifolds).
(2) A solution to the Bouchitté conjecture on optimal light structures and the justification of Michell trusses in engineering.
(3) The first rigorous results on fully nonlinear elasto-plastic evolution driven by the motion of (discrete) dislocations.
(4) Several partial results on quantitative rigidity of PDE-constrained measures, including applications to L^1 compensated compactness.
(5) The characterization of A-free measures by duality.
Altogether, a third of the problems outlined in the original proposal have been solved completely, another third has seen very substantial progress, and there are at least some partial results for the last third. Furthermore, deeper than anticipated connections to the theory of elasto-plasticity were discovered, which has led to a number of further questions to be investigated in follow-on research.
The most important results of the project are the following:
(1) Strong rectifiability results for PDE-constrained measures, providing a unified and generalized approach to a number of central rectifiability questions in geometric measure theory (e.g. for BV, BD, varifolds).
(2) A solution to the Bouchitté conjecture on optimal light structures and the justification of Michell trusses in engineering.
(3) The first rigorous results on fully nonlinear elasto-plastic evolution driven by the motion of (discrete) dislocations.
(4) Several partial results on quantitative rigidity of PDE-constrained measures, including applications to L^1 compensated compactness.
(5) The characterization of A-free measures by duality.
Altogether, a third of the problems outlined in the original proposal have been solved completely, another third has seen very substantial progress, and there are at least some partial results for the last third. Furthermore, deeper than anticipated connections to the theory of elasto-plasticity were discovered, which has led to a number of further questions to be investigated in follow-on research.
Many results beyond the state of the art at the beginning of the project have been discovered, in the form of both concrete results and novel techniques.
One particular accomplishment to be highlighted is the result that PDE-constrained measures have a certain dimensionality, in the sense that they are absolutely continuous with respect to a Hausdorff measure of suitable dimension. In fact, we even show that they are rectifiable of the same dimensions (where the dimensional upper density is positive). This is a much stronger result and allows one to prove a very large variety of existing rectifiability results within a unified framework (including generalized versions). The key idea is to show that the measure in question is not just absolutely continuous with respect to a Hausdorff measure of a certain dimension, but also with respect to the integral-geometric measure of the same dimension.
The solution to Bouchitté's optimal shape conjecture yielded a new compensated compactness technique, which combines Tartar’s compensated compactness arguments with careful addition of A-quasiconvex integrands and then pointwise optimizing them. While some ideas in this direction were known, they had not before been used for sequences merely converging weakly* in BV, BD or other spaces of A-free measures. The resulting theorem is important for applications, for the first time justifying fully the (modified) Michell truss approach to shape optimization problems in engineering, but also for improving our theoretical understanding of so-called "diffuse concentrations", which are at the heart of a number of applied and theoretical open problems in mathematical analysis.
Furthermore, in the course of this research, we have also discovered a number of unexpected connections, such as the convergence of a damage model to an elasto-plasticity model in the limit of vanishing strength of the damaged zone, and the importance of concentrations in elasto-plasticity theory.
In conclusion, the SINGULARITY project has contributed both to theoretical as well as applied problems and given much inspiration for future research directions.
One particular accomplishment to be highlighted is the result that PDE-constrained measures have a certain dimensionality, in the sense that they are absolutely continuous with respect to a Hausdorff measure of suitable dimension. In fact, we even show that they are rectifiable of the same dimensions (where the dimensional upper density is positive). This is a much stronger result and allows one to prove a very large variety of existing rectifiability results within a unified framework (including generalized versions). The key idea is to show that the measure in question is not just absolutely continuous with respect to a Hausdorff measure of a certain dimension, but also with respect to the integral-geometric measure of the same dimension.
The solution to Bouchitté's optimal shape conjecture yielded a new compensated compactness technique, which combines Tartar’s compensated compactness arguments with careful addition of A-quasiconvex integrands and then pointwise optimizing them. While some ideas in this direction were known, they had not before been used for sequences merely converging weakly* in BV, BD or other spaces of A-free measures. The resulting theorem is important for applications, for the first time justifying fully the (modified) Michell truss approach to shape optimization problems in engineering, but also for improving our theoretical understanding of so-called "diffuse concentrations", which are at the heart of a number of applied and theoretical open problems in mathematical analysis.
Furthermore, in the course of this research, we have also discovered a number of unexpected connections, such as the convergence of a damage model to an elasto-plasticity model in the limit of vanishing strength of the damaged zone, and the importance of concentrations in elasto-plasticity theory.
In conclusion, the SINGULARITY project has contributed both to theoretical as well as applied problems and given much inspiration for future research directions.