Abstract

In this talk we will deal with the characterization of the free boundary of the solutions to the following spectral $k$-partition problem with measure and inclusion constraints:    \begin{equation}\inf \left\{\sum_{i=1}^{k \lambda_1(\omega_i)\; : \;\begin{array}{c}\omega_i \subset \Omega \mbox{ are nonempty open sets for all } i=1,\ldots, k, \\ \omega_i \cap \omega_j = \emptyset \, \: \text{for all}\: i \not=j \mbox{ and } \sum_{i=1}}{k}|\omega_i| =  a\\\end{array}\right\},\end{equation}where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $a\in (0,|\Omega|)$. In particular, we will show free boundary optimality conditions, classify contact points, characterize the regular and singular part of the free boundary (including branching points), and describe the interaction of the partition with the fixed boundary $\partial \Omega$. The proof is based on a perturbed version of the problem, combined with monotonicity formulas, blowup analysis and classification of blowups, suitable deformations of optimal sets and eigenfunctions, as well as the improvement of flatness of [Russ-Trey-Velichkov, CVPDE 58 , 2019] for the one-phase points, and of [De Philippis-Spolaor-Velichkov, Invent. Math. 225, 2021] at two-phase points. Finally, if time permits, we will discuss some related problems that we are currently investigating, in particular the case when each component of the partition has a different measure constraint. This is a joint project with Makson S. Santos (Univ. Lisbon) and Hugo Tavares (IST Lisbon).