Mathematics Colloquium Series: Forcillo, Nicolò - Regularity issues in free boundary problems
2:45 p.m. to 3:45 p.m.
Date/Time: Monday 3/24, 2:45-3:45pm
Date/Time: Monday 3/24, 2:45-3:45pm
Location: FAB 1146
Speaker: Forcillo, Nicolò (Michigan State University)
Title: Regularity issues in free boundary problems
Abstract: In this talk, we introduce the topic of free boundary problems, focusing especially on regularity issues. Free boundary problems are a particular type of boundary value problem. They involve partial differential equations satisfied, in some sense, in domains depending on the solutions of the equations themselves. This is the peculiarity of this type of problems, since, generally, domains in boundary value problems are given as data. Additional conditions are imposed on a set determined by the boundaries of such domains. This set is the so-called free boundary of a solution to the free boundary problem.
The free boundary is an unknown object of the problem, at the same level as the solution itself. Its properties can be studied together or in parallel with the solution ones. Actually, there is a wide literature concerning the regularity of both the solution and the free boundary. In this context, a fundamental contribution had been given by the brilliant work of L. A. Caffarelli, paving the way for the investigation of free boundary problems in the variational framework. The optimal Lipschitz continuity and some regularity properties for the free boundary had been established for minimizers of one-phase and two-phase Bernoulli-type functionals. We remark that free boundary problems can be distinguished in two-phase and one-phase problems depending on whether the solution may change its sign or not respectively. A little bit later, Caffarelli had also developed a viscosity approach to deal with the Euler-Lagrange equation associated to these functionals (and generalizations), which turns out to be a free boundary problem involving the Laplacian. Caffarelli's use of viscosity methods for free boundary problems had been a breakthrough in the comprehension of them. After his works had appeared, many papers had been written, extending his contributions to more general operators, both in the variational and viscosity framework.
Another crucial contribution came from D. De Silva. Indeed, she provided an alternative method than Caffarelli's one to study the regularity of "flat" free boundaries for viscosity solutions to nonhomogeneous problems of the type of the ones studied by Caffarelli. Specifically, her approach relies on an ``improvement of flatness" result which, roughly speaking, says that if the graph of a solution oscillates ε-away from a hyperplane (ε-flat) in B1, the ball of radius 1 and center 0 in Rn, then it oscillates εr/2-away from possibly a different hyperplane in Br, for r sufficiently small. The key step of Caffarelli's method, instead, lies in finding a family of comparison subsolutions by using supconvolutions on balls of variable radii. The main advantage of De Silva's techniques compared to Caffarelli's approach, is that, up to now, it can be exploited to deal with nonhomogeneous problems employing viscosity techniques. The flexibility of her method allows us to study minimizers of Bernoulli-type functionals even using variational techniques and face thin free boundary problems as well.
In this talk, after going over these elements, we discuss some more recent objects of investigation in the theory of free boundary problems, the almost minimizers. In particular, we focus on the contribution I gave with S. Dipierro, F. Ferrari, and E. Valdinoci on almost minimizers for the one-phase p-Bernoulli functional. In this paper, we prove their optimal Lipschitz regularity when p>max{2n/(n+2),1}, taking inspiration from a paper by De Silva and O.Savin for p=2. We want to point out that there, remarkably, they were able to apply the techniques mentioned before by De Silva to explore the free boundary regularity for almost minimizers, which do not satisfy an equation in a variational sense (and so also in a viscosity sense). Going back to our work, our method mostly relies on using p-harmonic replacements as competitors. The regularity properties of these replacements allow us to infer the Lipschitz continuity of almost minimizes.