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.
