Advanced Course on Free Boundary Problems
to October 30, 2026
Venue: Centre de Recerca Matemàtica (CRM)
Room: Auditorium
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DESCRIPTION
This Advanced Course is part of the Intensive Research Programme on Analysis of Free Boundary Problems
LECTURERS
The two-phase Bernoulli problem: regularity and fine structure of the free boundaries
Bozhidar Velichkov
Università di Pisa
Abstract
Max Engelstein
University of Minnesota
One-phase free boundary problem and singularity analysi
Zihui Zhao
Johns Hopkins University
Abstract
scientific committee
Max Engelstein | University of Minnesota
Mariana Smit Vega Garcia | Western Washington University
Zihui Zhao | Johns Hopkins University
ORGANISING committee
Xavier Fernandez-Real | École polytechnique fédérale de Lausanne
Cole Jeznach | Universitat Autònoma de Barcelona
Xavier Ros Oton | ICREA – Universitat de Barcelona – Centre de Recerca Matemàtica
Xavier Tolsa | ICREA – Universitat Autònoma de Barcelona – Centre de Recerca Matemàtica
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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
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.