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

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• Dario Cesare Severo Mazzoleni (Università degli Studi di Pavia)
• Tuesday 03 February 2026, 15:50-16:30
• Seminar Room 1, Newton Institute.

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GSTW05 - Emerging Horizons in Geometric Spectral Theory: an ECRs workshop

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:

\[ \inf \left\{ \sum_{i=1}^{k \lambda_1(\omega_i)\; : \; \begin{array}{c} \omega_i \subset \Omega \text{ are nonempty open sets for all } i=1,\ldots, k, \\[0.2em] \omega_i \cap \omega_j = \emptyset \text{ for all } i \neq j \text{ and } \sum_{i=1}}{k} |\omega_i| = a \end{array} \right\}, \] where $\Omega$ is a bounded domain of $\mathbb{R}^N$ and $a \in (0, |\Omega|)$. 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, blow-up analysis and classification of blow-ups, 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, we will discuss some related problems that we are currently investigating. This is a joint project with Makson S. Santos (Univ. Lisbon) and Hugo Tavares (IST Lisbon).

This talk is part of the Isaac Newton Institute Seminar Series series.

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