Periodic Reporting for period 4 - EllipticPDE (Regularity and singularities in elliptic PDE's: beyond monotonicity formulas)
Período documentado: 2023-09-01 hasta 2024-09-30
Broadly speaking, the goal of this project is to develop substantially the regularity theory for elliptic Partial Differential Equations (PDE).
PDE are used in essentially all sciences and engineering, and have important connections with several branches of pure mathematics.
The question of regularity is one of the most basic and important mathematical questions in PDE theory: to understand whether all solutions to a given PDE are smooth or if, instead, they may have singularities.
The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context.
In particular, three key objectives of the project are the following.
First, to introduce new techniques to obtain a deeper understanding of singularities in nonlinear elliptic PDE. Aside from its intrinsic interest, this is likely to provide insightful applications in other contexts.
A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. In other words, to prove that 'almost every' solution is smooth.
Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE that does not rely on monotonicity formulas.
These three objectives, while seemingly different, are in fact deeply interrelated.
PDE are used in essentially all sciences and engineering, and have important connections with several branches of pure mathematics.
The question of regularity is one of the most basic and important mathematical questions in PDE theory: to understand whether all solutions to a given PDE are smooth or if, instead, they may have singularities.
The aim of this project is to go significantly beyond the state of the art in some of the most important open questions in this context.
In particular, three key objectives of the project are the following.
First, to introduce new techniques to obtain a deeper understanding of singularities in nonlinear elliptic PDE. Aside from its intrinsic interest, this is likely to provide insightful applications in other contexts.
A second aim of the project is to establish generic regularity results for free boundaries and other PDE problems. In other words, to prove that 'almost every' solution is smooth.
Finally, the third main objective is to achieve a complete regularity theory for nonlinear elliptic PDE that does not rely on monotonicity formulas.
These three objectives, while seemingly different, are in fact deeply interrelated.
Recall that free boundary problems are those described by PDE that exhibit a priori unknown (free) interfaces or boundaries. These problems appear in Probability, Physics, Biology, Finance, or Industry, and the most classical example is the melting of ice (phase transition).
This project has produced various grounbreaking results in the theory of free boundary problems (FBP), as explained next.
The oldest and most important FBP is the Stefan problem, a PDE which describes mathematically the melting of ice, and was first introduced in 1831. In this context, it was known that the ice-water interface could develop singularities, but their understanding was still quite limited. In a groundbreaking work [J. Amer. Math. Soc. 2024], the PI and his collaborators proved that, while singularities may appear, the set of all singularities is small. Their precise result (whose proof is more than 80 pages long and had to develop new techniques in Geometric Measure Theory) completely solved a long-standing conjecture, giving for the first time a complete understanding of this model that dates back to the 19th century. It is the best result for the Stefan problem since the famous work of Caffarelli [Acta Math. 1977], one of the main results for which he got the Abel Prize.
Another key result that has been achieved so far is the understanding of the generic regularity of free boundaries in the so-called obstacle problem. In this context, known examples showed that some solutions exhibit many singularities (i.e. the singular set is not small). Still, these solutions were expected to be rare. This is exactly what the PI and his collaborators proved in [Publ. Math. IHÉS 2020], solving a conjecture of Schaeffer (dating back to 1974) that states that free boundaries are smooth for 'almost every' solution.
On the other hand, another goal of the project that has been achieved is the optimal regularity of solutions to the fully nonlinear obstacle problem [JEMS 2024] and the higher regularity of free boundaries in problems with nonlocal interactions [Adv. Math. 2020]. These were important open problems, and the PI has solved them by introducing new techniques that have already find applications in other contexts.
These outstanding achievements have been disseminated, both inside the mathematical community and to the general public.
For the mathematical community, the PI has written two books [EMS Press 2022] and [Birkhäuser 2024], and has given several courses for young researchers at summer schools and workshops. For the general public, the PI has given public talks on several occasions, and Quanta Magazine (one of the best science dissemination journals, founded by the Simons Foundation) published an outreach article about his work on the Stefan problem.
This project has produced various grounbreaking results in the theory of free boundary problems (FBP), as explained next.
The oldest and most important FBP is the Stefan problem, a PDE which describes mathematically the melting of ice, and was first introduced in 1831. In this context, it was known that the ice-water interface could develop singularities, but their understanding was still quite limited. In a groundbreaking work [J. Amer. Math. Soc. 2024], the PI and his collaborators proved that, while singularities may appear, the set of all singularities is small. Their precise result (whose proof is more than 80 pages long and had to develop new techniques in Geometric Measure Theory) completely solved a long-standing conjecture, giving for the first time a complete understanding of this model that dates back to the 19th century. It is the best result for the Stefan problem since the famous work of Caffarelli [Acta Math. 1977], one of the main results for which he got the Abel Prize.
Another key result that has been achieved so far is the understanding of the generic regularity of free boundaries in the so-called obstacle problem. In this context, known examples showed that some solutions exhibit many singularities (i.e. the singular set is not small). Still, these solutions were expected to be rare. This is exactly what the PI and his collaborators proved in [Publ. Math. IHÉS 2020], solving a conjecture of Schaeffer (dating back to 1974) that states that free boundaries are smooth for 'almost every' solution.
On the other hand, another goal of the project that has been achieved is the optimal regularity of solutions to the fully nonlinear obstacle problem [JEMS 2024] and the higher regularity of free boundaries in problems with nonlocal interactions [Adv. Math. 2020]. These were important open problems, and the PI has solved them by introducing new techniques that have already find applications in other contexts.
These outstanding achievements have been disseminated, both inside the mathematical community and to the general public.
For the mathematical community, the PI has written two books [EMS Press 2022] and [Birkhäuser 2024], and has given several courses for young researchers at summer schools and workshops. For the general public, the PI has given public talks on several occasions, and Quanta Magazine (one of the best science dissemination journals, founded by the Simons Foundation) published an outreach article about his work on the Stefan problem.
The understanding of generic regularity is one of the main challenges in the whole field of nonlinear PDEs.
Two groundbreaking achievements of this project, which go significantly beyond the state of the art in this context, is our recent solution of Schaeffer's conjecture, as well as our complete understanding of the singular set in the Stefan problem.
These important breakthroughs have been published in Publ. Math. IHES and JAMS, two of the top journals in mathematics, and will yield new developments and applications in other contexts.
Two groundbreaking achievements of this project, which go significantly beyond the state of the art in this context, is our recent solution of Schaeffer's conjecture, as well as our complete understanding of the singular set in the Stefan problem.
These important breakthroughs have been published in Publ. Math. IHES and JAMS, two of the top journals in mathematics, and will yield new developments and applications in other contexts.