This course is dedicated to the two-phase Bernoulli free boundary problem. I will discuss the theory in the general case, in which the solution changes sign and might have zero level set of positive Lebesgue measure. Alt, Caffarelli and Friedman formulated the problem in the 80’s and proved the Lipschitz continuity of the solutions by introducing the Alt-Caffarelli-Friedman’s monotonicity formula. I obtained the regularity of the free boundaries of the positive and the negative phases with Spolaor in 2016 via an epiperimetric inequality in 2D, while with De Philippis and Spolaor, we solved the higher dimensional case in 2021 via an improvement of flatness argument à la De Silva. I will also discuss the structure of the contact set between the free boundaries of the positive and the negative parts. Precisely, I will show that in 2D the contact set is locally composed of finitely many disjoint smooth arcs. The proof is based on a Weierstrass-type representation formula, inspired by the works of Hélein-Hauswirth-Pacard and Traizet, which allows to rewrite the problem into a geometric free boundary problem for minimal surfaces in the half 3D space.

Supercooling is the process of cooling a liquid to below its freezing temperature yet maintaining its liquid state (due to certain chemical properties of the liquid molecules). This configuration is quite unstable and when perturbed the liquid freezes in unpredictable and pathological ways. Ice can appear out of “nowhere” (nucleation), fractal shapes develop in the liquid ice interface and freezing fronts can move with “infinite speed”. You can see these phenomena at home with your freezer (or on YouTube).

In this mini-course we will discuss a mathematical model for this physical process called the Supercooled Stefan Problem. This is an ill-posed parabolic free boundary problem. We are most interested in instances when the problem can be regularized, giving rise to the parabolic obstacle problem

Interestingly, this regularized model exhibits some of the pathological behavior of freezing supercooled liquids (nucleation and rapidly advancing freezing fronts) but provably does not develop large fractal liquid-ice interfaces generically. We will present this proof, which is joint work with I. Kim (UCLA) and S. Munoz (UCLA). Along the way we will discuss other work on parabolic obstacle problems, especially that of Caffarelli, Petrosyan and Shahgholian (JAMS 2004) and Figalli, Ros-Oton and Serra (JAMS 2024), both of which majorly influence and inform the analysis of the supercooled problem.

The topic of the mini-course is the one-phase free boundary problem, namely the variational problem of the Alt-Caffarelli energy functional
subject to prescribed boundary condition. As is often the case in free boundary problems, the regularity of solutions is tied to the regularity of the corresponding free boundaries. To study the latter, we perform a blow-up analysis and show that singular points of the free boundary are locally modeled on non-flat one-homogeneous solutions to the problem. We discuss a few examples of these homogeneous solutions. Further analysis near these tangent functions reveals the structure of the free boundary near singularity.