Regularity and Stability in Partial Differential Equations

Project Information

RSPDE

Grant agreement ID: 721675

DOI

10.3030/721675

Project closed

EC signature date 2 December 2016

Start date 1 February 2017

End date 31 July 2023

Funded under

• EXCELLENT SCIENCE - European Research Council (ERC)

Total cost

€ 1 742 428,00

EU contribution

€ 1 742 428,00

Coordinated by

EIDGENOESSISCHE TECHNISCHE HOCHSCHULE ZUERICH
Switzerland

Periodic Reporting for period 4 - RSPDE (Regularity and Stability in Partial Differential Equations)

Reporting period: 2021-08-01 to 2023-07-31

Summary of the context and overall objectives of the project

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.

Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far

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.

Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far)

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)

ERC_RSPDE