Free boundary problems are those described by PDEs that exhibit a priori unknown (free) interfaces or boundaries. In other words, there are two unknowns: the solution of a PDE, and a free boundary which determines the domain in which the PDE is satisfied. These problems appear in Probability, Physics, Biology, Finance, or Industry, and the study of solutions and free boundaries uses methods from PDEs, Calculus of Variations, and Geometric Measure Theory. The main mathematical challenge is understanding the regularity of free boundaries.
The obstacle problem is the most classical and motivating example in the study of free boundary problems. A milestone in this context is a classical paper of L. Caffarelli (Acta Math. 1977), in which he established for the first time the regularity of free boundaries in the obstacle problem, outside a certain set of singular points. In a recent work with Figalli and Serra we proved that, generically, the singular set has zero (n-4)-dimensional Hausdorff measure (Publ. Math. IHÉS 2020). In particular, this solved a long-standing conjecture of Schaeffer (dating back to 1974) in dimensions n≤4.
On the other hand, motivated by the applications mentioned above, in the last decades there has been a growing interest in understanding thin obstacle problems (in which the free boundary is lower-dimensional), and obstacle problems for integro-differential operators (in which the equation is non-local). One of my most important contributions in this context is a joint work with L. Caffarelli and J. Serra (Invent. Math. 2017), in which we established the regularity of free boundaries in obstacle problems for integro-differential operators. This was an important and long-standing question in the field. Moreover, in collaboration with B. Barrios and A. Figalli, we proved several general results regarding the regularity of free boundaries in the elliptic and parabolic fractional obstacle problems (Amer. J. Math. 2018 & Comm. Pure Appl. Math. 2018).
PDEs are relations between an unknown function and its partial derivatives. Integro-differential equations are of similar nature, but involve operators that are non-local (i.e., to evaluate the operator at a point, information about the function far from that point is needed). Such kind of equations arise naturally in the study of stochastic processes with jumps: Lévy processes. For example, the study of such processes leads naturally to linear problems posed in bounded domains. Furthermore, stochastic control problems involving these jump-diffusions lead to nonlinear integro-differential equations.
The study of integro-differential equations is a broad field, in which I have made important contributions in the last years. A central problem in this context is to understand the regularity of solutions to elliptic and parabolic integro-differential equations. The regularity theory for such equations started already in the fifties and sixties, but has been mostly developed in the last 20 years. It is currently a very active field of research, with several interesting open questions. One of my most important contributions in this context is a joint work with J. Serra (Duke Math. J. 2016), in which we developed the boundary regularity theory for fully nonlinear integro-differential elliptic equations.
Many PDEs can be written as the Euler-Lagrange equation of an appropriate functional. In some cases, existence and uniqueness of solutions to the PDE follows by showing that the functional is uniformly convex and has a global minimizer. In other cases, however, there may be multiple solutions to the PDE (corresponding to multiple critical points of the functional), and the situation is more delicate.
My research in this context concerns the regularity of local minimizers and, more generally, of stable solutions. This is a classical question in the Calculus of Variations. An important example in Geometry is the regularity of minimal hypersurfaces of R^n which are minimizers of the area functional. A deep result from the seventies states that these hypersurfaces are smooth if n≤7, while in R^8 the Simons cone is a minimizer of the area with a singularity at the origin.
The same phenomenon --the fact that regularity holds in low dimensions-- happens for reaction-diffusion problems in bounded domains. A long standing conjecture in this context was to prove that stable solutions to such problems are always bounded (and hence smooth) in dimensions n≤9. We completely solved this conjecture in a joint work with X. Cabré, A. Figalli, and J. Serra (Acta Math. 2020).
Fully nonlinear elliptic equations are highly nonlinear PDEs. They appear in Probability (stochastic control or differential games) as well as in Geometry. These are non-variational equations (i.e., there is no functional associated to the equation), and to prove existence and uniqueness of solutions one needs to focus attention on viscosity solutions, which are defined via the comparison principle. The key challenge in the study of such equations is to understand the regularity of solutions. For example, it is an outstanding open problem to decide wether all solutions to fully nonlinear elliptic equations in R^3 are smooth. The result is known to be true in R^2, while in dimensions 5 and higher there are counterexamples.
Evolution equations are PDEs that model diffusion phenomena. The quantity that diffuses can be a concentration, heat, momentum, a population, or a price; every such process is governed by a PDE. There are many different types of evolution equations, the simplest one is the classical heat equation. Some of these equations are nonlinear, some of them are non-local, and in some cases they even develop free boundaries. Important questions in this context are the regularity of solutions, their long time behavior, finite vs infinite speed of propagation, etc.
The isoperimetric problem consists of characterizing minimizers of the perimeter among those sets having volume equal to a certain constant. Isoperimetric-type inequalities play an important role in Analysis and in the study of PDEs, and have also applications in Probability and Geometry. They are flexible tools and are useful in many different settings. Together with X. Cabré and J. Serra (J. Eur. Math. Soc. 2016), we used techniques coming from the theory of fully nonlinear equations to obtain a whole new family of sharp isoperimetric inequalities with weights. As a particular case of our results, we gave new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.
The existence of periodic solutions in one-dimensional ODEs is a natural and important question, which is related to Hilbert 16th problem. The family of Abel equations of the second kind is of special mathematical interest because, while it appears frequently in applied mathematics, the equation becomes singular whenever the solution vanishes. Thus, usual existence/uniqueness ODE techniques can not be used in this context, and the understanding of periodic solutions (with non-constant sign) becomes a much more delicate issue.