Periodic Reporting for period 3 - EllipticPDE (Regularity and singularities in elliptic PDE's: beyond monotonicity formulas)
Reporting period: 2022-03-01 to 2023-08-31
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).
A key result that has been achieved so far is the understanding of the generic regularity of free boundaries in the so-called obstacle problem.
More precisely, we have solved a conjecture of Schaeffer (dating back to 1974), proving for the first time that free boundaries actually are smooth for 'almost every' solution.
This work has already been accepted for publication in Publ. Math. IHES, one of the top journals in mathematics.
Furthermore, using the ideas introduced in such work we have been able to establish an analogous result for the thin obstacle problem; a classical model that appears in elasticity and semipermeable membranes.
We expect these ideas to be applicable in other contexts, too.
On the other hand, another goal of the project that has been achieved is the higher regularity of free boundaries in problems with nonlocal interactions.
This was an open problem, and we have solved it by introducing new techniques that we will pursue further in the near future.
A key result that has been achieved so far is the understanding of the generic regularity of free boundaries in the so-called obstacle problem.
More precisely, we have solved a conjecture of Schaeffer (dating back to 1974), proving for the first time that free boundaries actually are smooth for 'almost every' solution.
This work has already been accepted for publication in Publ. Math. IHES, one of the top journals in mathematics.
Furthermore, using the ideas introduced in such work we have been able to establish an analogous result for the thin obstacle problem; a classical model that appears in elasticity and semipermeable membranes.
We expect these ideas to be applicable in other contexts, too.
On the other hand, another goal of the project that has been achieved is the higher regularity of free boundaries in problems with nonlocal interactions.
This was an open problem, and we have solved it by introducing new techniques that we will pursue further in the near future.
The understanding of generic regularity is one of the main challenges in the whole field of nonlinear PDEs.
An important achievement of this project, which goes significantly beyond the state of the art in this context, is our recent solution of Schaeffer's conjecture.
This important breakthrough has been published in Publ. Math. IHES, one of the top journals in mathematics, and will yield new developments and applications in other contexts.
Indeed, the techniques introduced in such work will be applied in case of the Stefan problem (which models phase transitions), and we expect to obtain a much deeper understanding of the singular set of the Stefan problem. This is a classical model from the XIXth century, for which not much is known about its corresponding set of singularities.
On the other hand, we also expect our work on generic regularity for the obstacle problem to yield a new dichotomy for singular points in all dimensions, extending previous results of Sakai in 2D.
An important achievement of this project, which goes significantly beyond the state of the art in this context, is our recent solution of Schaeffer's conjecture.
This important breakthrough has been published in Publ. Math. IHES, one of the top journals in mathematics, and will yield new developments and applications in other contexts.
Indeed, the techniques introduced in such work will be applied in case of the Stefan problem (which models phase transitions), and we expect to obtain a much deeper understanding of the singular set of the Stefan problem. This is a classical model from the XIXth century, for which not much is known about its corresponding set of singularities.
On the other hand, we also expect our work on generic regularity for the obstacle problem to yield a new dichotomy for singular points in all dimensions, extending previous results of Sakai in 2D.