Periodic Reporting for period 4 - RSPDE (Regularity and Stability in Partial Differential Equations)
Periodo di rendicontazione: 2021-08-01 al 2023-07-31
This project used several deep techniques in mathematics to progress in a series of fundamental problems.
These are:
1) Optimal transport: what is the cheapest way to transport resources from one place to another? We recall that optimal transport is a fundamental tool in many applications to geometry and PDEs.
2) Functional inequalities: these mathematical tools allow one to study/understand equilibrium configurations for crystals and for several important dynamical evolution problems.
3) Stability in PDEs: this broad question can be used to understand the behavior of physical phenomena where solutions exhibit singularities or free boundaries.
These questions/problems are mathematically related, and their understanding allows one to have better knowledge of the physical systems that they aim to describe.
In these years we managed to obtain several fundamental advancements in these areas.
These are:
1) Optimal transport: what is the cheapest way to transport resources from one place to another? We recall that optimal transport is a fundamental tool in many applications to geometry and PDEs.
2) Functional inequalities: these mathematical tools allow one to study/understand equilibrium configurations for crystals and for several important dynamical evolution problems.
3) Stability in PDEs: this broad question can be used to understand the behavior of physical phenomena where solutions exhibit singularities or free boundaries.
These questions/problems are mathematically related, and their understanding allows one to have better knowledge of the physical systems that they aim to describe.
In these years we managed to obtain several fundamental advancements in these areas.
The group has obtained a series of breakthroughs in all the problems described above, as described in the section "major achievements".
By introducing new tools and ideas, we have obtained several new stability results, both for functional and geometric inequalities (e.g. Sobolev inequalities) and in the field of free boundary problems.
Also the theory of optimal transport has been very successful: we obtained regularity estimates for optimal maps between unbounded domains (this is very important in applications) and we also wrote a new monograph on the topic, published by the EMS.
By introducing new tools and ideas, we have obtained several new stability results, both for functional and geometric inequalities (e.g. Sobolev inequalities) and in the field of free boundary problems.
Also the theory of optimal transport has been very successful: we obtained regularity estimates for optimal maps between unbounded domains (this is very important in applications) and we also wrote a new monograph on the topic, published by the EMS.
We have solved a series of major conjectures in elliptic PDEs and free boundaries, more notably:
- Brezis conjecture on stable solutions to semilinear elliptic PDEs (Acta Math 2020)
- Schaeffer conjecture in dimensions at most 4 on the generic regularity of the free boundary in the obstacle problem (Publ IHES 2020)
- a variant of the De Giorgi conjecture for the fractional laplacian in dimension 5 (Invent Math 2019)
- Brezis conjecture on stable solutions to semilinear elliptic PDEs (Acta Math 2020)
- Schaeffer conjecture in dimensions at most 4 on the generic regularity of the free boundary in the obstacle problem (Publ IHES 2020)
- a variant of the De Giorgi conjecture for the fractional laplacian in dimension 5 (Invent Math 2019)