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Free boundary regularity for a spectral optimal partition problem with volume and inclusion constraints

This paper is devoted to a complete characterization of the free boundary of all solutions to the following spectral $k$-partition problem with measure and inclusion constraints: \[ \inf \left\{\sum_{i=1}^k λ_1(ω_i)\; : \; ω_i \subset Ω\mbox{ are nonempty open sets for all } i=1,\ldots, k,\; ω_i \cap ω_j = \emptyset \: \text{for all}\: i \not=j \mbox{ and } \sum_{i=1}^{k}|ω_i| = a \right\}, \] where $Ω$ is a bounded domain of $\mathbb{R}^N$, $a\in (0,|Ω|)$. 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 $\partial Ω$. 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.