*Preprint*

**Inserted:** 23 sep 2024

**Last Updated:** 23 sep 2024

**Year:** 2024

**Abstract:**

This paper is devoted to a complete characterization of the free boundary of a particular solution to the following spectral $k$-partition problem with measure and inclusion constraints: \[ \inf \left\{\sum_{i=1}^k \lambda_1(\omega_i)\; : \; \omega_i \subset \Omega \mbox{ are nonempty open sets for all } i=1,\ldots, k, \; \omega_i \cap \omega_j = \emptyset \: \text{for all}\: i \not=j \mbox{ and } \sum_{i=1}^{k} \mathcal L^N(\omega_i ) = a \right\}, \] where $\Omega$ is a bounded domain of $\mathbb{R}^N$, $a\in (0,\mathcal L^N(\Omega))$. In particular, we prove free boundary 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.

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