Variational Methods for a Two-Phase Free Boundary Problem for Harmonic Measure

Spiraling asymptotic profiles of competition-diffusion systems

Minimization of the compliance of a connected 1-dimensional set

Optimal regularity for three dimensional mass minimizing cones in arbitrary codimension

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Max Engelstein:

Variational Methods for a Two-Phase Free Boundary Problem for Harmonic Measure

We study the regularity and structure of the singular set of a two-phase free boundary problem for harmonic measure. To do so, we combine "variational tools", such as monotonicity formulas and an epiperimetric inequality, with estimates from harmonic analysis and geometric measure theory. Some of this is joint work with Matthew Badger (University of Connecticut) and Tatiana Toro (University of Washington).

Jimmy Lamboley:

Minimization of the compliance of a connected 1-dimensional set

In this talk, we describe the features of the following optimization problem, whose unknown is a connected one-dimensional set in the plane:

$\mathrm{min}$ { $\mathcal{C}\left(\Sigma \right)+\lambda {\mathscr{H}}^{1}\left(\Sigma \right)$ , $\Sigma $ closed connected subset of $\overline{\Omega}$ },

where $\Omega $ is a fixed open set of the plane, $\lambda >0$, ${\mathscr{H}}^{1}\left(\Sigma \right)$ denotes the length of $\Sigma $ (its 1-dimensional Hausdorff measure), and $\mathcal{C}\left(\Sigma \right)$ denotes the compliance of $\Omega \setminus \Sigma $, that is the opposite of the Dirichlet energy of $\Omega \setminus \Sigma $ (for an external force term $f\in {L}^{2}\left(\Omega \right)$). This problem can be interpreted as to find the best location for attaching (on $\Sigma $) a membrane subject to a given external force $f$ so as to minimize its compliance. It can be seen as an elliptic PDE version of the average distance problem/irrigation problem which has been extensively studied in the literature, and is also deeply related to the famous Mumford-Shah problem. We particularly focus on the regularity and the topology of minimizers, we prove that they are made of a finite number of smooth curves meeting only by three at 120 degree angles, containing no loop, and possibly touching $\partial \Omega $ only tangentially. We will describe the classical tools and the new ones we developed for this purpose. This is a joint work with A. Chambolle, A. Lemenant and E. Stepanov.

Alessandro Zilio:

Variational Methods for a Two-Phase Free Boundary Problem for Harmonic Measure

We consider solutions of a two dimensional competitive Lotka-Volterra model, described by the elliptic system

$-\Delta {u}_{i}=-\beta {u}_{i}{\displaystyle \sum _{j\ne i}}{a}_{ij}{u}_{j}\phantom{\rule{2.00em}{0ex}}({a}_{ij}>0)$

This prototypical system is widely used to model competition between biological species, and is also related to some interesting problems in optimal partition problems. In this talk, I will focus on the asymmetric case ${a}_{ij}\ne {a}_{ji}$ and on the structure of the limit free-boundary problem that follows in the singular limit $\beta \to +\infty $. This is a joint result with A. Salort, S. Terracini and G. Verzini.

Luca Spolaor:

Optimal regularity for three dimensional mass minimizing cones in arbitrary codimension.

In this talk I will present the following regularity results: the singular set of a mass minimizing three dimensional cone in the Euclidean space is at most the union of finitely many half lines (joint with C. De Lellis and E. Spadaro).