Free Boundary Problems

WORKSHOP

Workshop on Free Boundary Problems

Workshop

From November 02, 2026
to November 06, 2026

Venue: Centre de Recerca Matemàtica (CRM)

Room: Auditorium

Registration deadline 12 / 10 / 2026

DESCRIPTION

This Workshop is part of the Intensive Research Programme on Analysis of Free Boundary Problems

LECTURERS

Harmonic measure, hausdorff content estimates, and sets of dimension 2.999999

Matthew Badger

University of Connecticut

Abstract

Existence of some orthogonal families of curves in a rectangle

Guy David

Université Paris-Saclay

Abstract

Recent Progress on the Parabolic David-Semmes Conjecture

John Hoffman

Florida State University

Abstract

Inwon Kim

University of California – Los Angeles

Non-minimizing and min-max solutions to Bernoulli problems

Dennis Kriventsov

Rutgers University

Abstract

Svitlana Mayboroda

ETH Zurich, University of Minnesota, IAS

Analysis vs. Geometry for the Regularity Problem for Elliptic Operators in Rough Domains

Mihalis Mourgoglou

Ikerbasque, EHU

Abstract

Robin Neumayer

Carnegie Mellon University

Multi-operator two-phase elliptic measure: regularity and structure

Anna Skorobogatova

Institute for Theoretical Studies / ETH Zurich

Abstract

Parabolic Theory as a High-Dimensional Limit of Elliptic Theory

Marianna Smit Vega Garcia

Western Washington University

Abstract

Luca Spolaor

University of California – San Diego

Singularly perturbed elliptic systems modeling partial separation and their free boundaries

Susanna Terracini

Università di Torino

Abstract

Eduardo Teixeira

Oklahoma State University

Quantitative differentiation, Poincarè inequalities and statistical learning theory

Michele Villa

Ikerbasque, EHU

Abstract

Slow decay vs. fast decay around minimal cones

Kelei Wang

Wuhan University

Abstract

Hui Yu

National University of Singapore

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

LIST OF PARTICIPANTS

Name	Institution

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Identifying the relationship between the structure of a given rough boundary and the behavior of its elliptic measure has a long history. In the two-phase setting, one typically wishes to deduce geometric information about the boundary from analytic information on the pair of elliptic measures from either side. In the multi-operator setting, the coefficients of the operators from either side may have a discontinuity across the boundary, and thus many of the techniques used in the single-operator setting are not applicable. I will discuss some regularity and structural results, analogous to those in the case of a single operator, which we are able to recover in spite of this. In particular, we will see how to get around the lack of available monotonicity formulas in this setting via improvement of flatness techniques for solutions of a suitable free boundary problem. This is based on joint work with Max Goering, and joint work with Cole Jeznach.

A central theme in modern harmonic analysis and elliptic PDE is the interplay between the analytic properties of boundary value problems and the geometry of the underlying domain. In this talk, I will discuss this principle in the context of the Regularity problem for divergence-form elliptic operators in rough domains.
On the one hand, quantitative geometric assumptions on the boundary, such as uniform rectifiability, imply solvability of the Regularity problem with Sobolev data. On the other hand, solvability itself carries geometric information. I will describe recent results showing that the solvability of the Regularity problem characterizes uniform rectifiability of the boundary, thus establishing an equivalence between analytic and geometric regularity.
I will also discuss a qualitative counterpart of this picture. In this setting, quantitative estimates are replaced by qualitative assumptions, and solvability leads naturally to rectifiability of the boundary. This perspective connects boundary value problems, harmonic measure, and geometric measure theory, and suggests new directions toward free boundary problems where geometry is recovered from analytic information.

In this talk I will give an overview of various recent results concerning quantitative differentiation of Lipschitz and Sobolev functions in geometric contexts. For example, I will present an extension of Dorronsoro theorem to uniformly rectifiable sets and show that all uniformly rectifiable sets may be covered with a surface supporting a Poincaré inequality. I will then apply this theory to problems in unsupervised and supervised statistical learning. These results come from various joint works with J. Azzam, E. Caputo, M. Hyde, M. Mourgoglou, R. Schul and I. Violo.

In the Euclidean setting, David and Semmes conjectured that, in the presence of the ADR condition, $\ L^2$ boundedness of the Riesz transform implies uniform rectifiability of the underlying measure. We present recent progress on the parabolic analog of this conjecture.

Bernoulli type free boundary problems have a well-developed existence and regularity theory. Much of this, however, is restricted to the case of minimizers of the natural energy (the Alt-Caffarelli functional). I will describe a compactness and regularity theorem that applies to any critical point instead, based on a nonlinear frequency formula and Naber-Valtorta estimates. Then I will explain, via an example involving gravity water waves, how to use this theorem to find min-max type (mountain pass) solutions. This is based on joint work with Georg Weiss.

This is joint work with Polina Perstneva, to be written (so, not yet secured but probably interesting anyway). The initial question concerned the construction, on domains of the plane bounded by general snowflakes, of elliptic operators with a given harmonic measure on the boundary. For this we like to construct the level sets of the Green function first, and the desired ellipticity of the constructed operator translates into a long distance regularity of the distance between the orthogonal gradient lines. A regulation tool is needed, which translates into the following control problem on the square. Can we find two mutually orthogonal sets of curves in the unit square (green and red), so that the green curves connect points $\ (0, t)$ on the left to to $\ (1, t)$ on the right, and the red ones go from $\ (x, 0) on the bottomto$ (f(x), 1) [/latex] on the top, where $f$ is any given small enough compact perturbation of the identity in $[0, 1]$ ?

The dimension $d_n$ of harmonic measure in $\mathbb{R}^n$ is the maximal minimum Hausdorff dimension of a subset of a boundary of a bounded domain through which the trace of Brownian motion first exits the domain almost surely. From theorems of Makarov, Jones, Wolff, and Bourgain in the 1980s, it is known that $d_2 = 1$ and $n - 1 < d_n < n$ for all $n \geq 3$ .

In the early 1990s, Bishop conjectured that $d_3 = 2.5$ and $d_n = n - \frac{1}{n - 1}$ for all $n \geq 3$ , but the conjecture is wide open. I will discuss ideas behind the state-of-the-art upper bounds $d_3 < 2.999999$ and $d_n < n - \left(\frac{1}{n - 1}\right)^{7n}$ for all $n \geq 3$ , including building covers with disjoint cubes in general position and a flexible scheme to approximate polar integration against Ahlfors regular measures.

I will also present several challenges on the dimension of harmonic measure that could plausibly be solved before Bishop’s conjecture. This talk is based on joint work with Michael Albert (University of Connecticut, Okinawa Institute of Science and Technology) and Alyssa Genschaw (Milwaukee School of Engineering).

We investigate the asymptotic behavior, as $\beta \to +\infty$ , of solutions to competition-diffusion systems of type

$\begin{cases} \Delta u_{i,\beta} = \beta u_{i,\beta} \prod_{j \ne i} u_{j,\beta}^{2} & \text{in } \Omega, \\ u_{i,\beta} = \varphi_i \ge 0 & \text{on } \partial\Omega, \end{cases} \quad i = 1,2,3 \,$

where $\varphi_i \in W^{1,\infty}(\Omega)$ satisfy the partial segregation condition

$\varphi_1 \varphi_2 \varphi_3 \equiv 0$ in $\overline{\Omega}$ .

For $\beta > 1$ fixed, a solution can be obtained as a minimizer of the functional

$\displaystyle J_\beta(\mathbf{u},\Omega) := \int_\Omega \left( \sum_{i=1}^{3} |\nabla u_i|^2 + \beta \prod_{j=1}^{3} u_j^2 \right) dx$

on the set of functions in $H^1(\Omega,\mathbb{R}^3)$ with fixed traces on $\partial\Omega$ .

We prove a priori and uniform (in $\beta$ ) Hölder bounds. In the limit, we are led to minimize the energy

$J(\mathbf{u},\Omega) := \int_\Omega \sum_{i=1}^3 |\nabla u_i|^2 dx$

over all partially segregated states

$u_1 u_2 u_3 \equiv 0$ in $\overline{\Omega}$

satisfying the given partially segregated boundary conditions above. We prove regularity of the free boundary up to a low-dimensional singular set.

Since elliptic PDE theory can be seen as a steady-state version of parabolic PDE theory, if a parabolic estimate holds, then by eliminating the time parameter, one can obtain an underlying elliptic statement. Producing a parabolic statement from an elliptic statement, on the other hand, is not as straightforward. In this talk, we will discuss how a high-dimensional limiting technique can be used to prove theorems about solutions to the fractional heat equation from their elliptic counterparts. This is a joint work with Blair Davey.

In the study of the Bernstein problem, Bombieri-De Giorgi-Giusti established the existence of minimal foliations around the Simons cone. Later, Hardt-Simon identified a remarkable condition, the strictly minimizing condition, which determines the decay rate of smooth minimal hypersurfaces in this foliation as they approach the minimal cone, i.e. if it is a slow decay or fast decay. This condition proves highly useful in numerous other problems, such as the construction of entire minimal graphs. In this talk, I will first provide a review of this slow decay vs. fast decay issue, and then discuss similar phenomena in several other problems, including the Allen-Cahn equation and the Alt-Caffarelli one-phase free boundary problem.