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.
