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

Dario Mazzoleni, Makson S. Santos, Hugo Tavares

TL;DR

This work characterizes the free boundary for optimal spectral $k$-partitions with volume constraints on a bounded domain, proving a full regularity/structure theorem that decomposes each boundary into interior one-phase, interior two-phase, and boundary-contact points with a regular/singular split inside $∂Ω_i$. The authors introduce a penalized problem $J_η$ to handle the volume constraint, establish existence, nondegeneracy, Lipschitz continuity, and finite perimeter for minimizers, and prove that triple intersections are forbidden while two-phase contacts at the fixed boundary do not occur. A central part of the analysis is a blow-up study at two-phase points, using Alt–Caffarelli–Friedman monotonicity to show blow-ups are two-plane solutions, and a viscosity-based optimality framework to transfer these local structures into global regularity; for small $η$, the penalized problem is equivalent to the original constrained problem, yielding a complete description of optimal partitions. The results extend the free boundary theory to spectral partitions with volume constraints, providing rigorous blow-up classifications, optimality conditions, and interaction with the domain boundary, with implications for the geometry and regularity of optimal partitions. Overall, the paper advances the understanding of spectral partition problems by integrating penalization techniques, monotonicity formulas, and viscosity methods to achieve sharp free boundary regularity results.

Abstract

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.

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

TL;DR

This work characterizes the free boundary for optimal spectral -partitions with volume constraints on a bounded domain, proving a full regularity/structure theorem that decomposes each boundary into interior one-phase, interior two-phase, and boundary-contact points with a regular/singular split inside . The authors introduce a penalized problem to handle the volume constraint, establish existence, nondegeneracy, Lipschitz continuity, and finite perimeter for minimizers, and prove that triple intersections are forbidden while two-phase contacts at the fixed boundary do not occur. A central part of the analysis is a blow-up study at two-phase points, using Alt–Caffarelli–Friedman monotonicity to show blow-ups are two-plane solutions, and a viscosity-based optimality framework to transfer these local structures into global regularity; for small , the penalized problem is equivalent to the original constrained problem, yielding a complete description of optimal partitions. The results extend the free boundary theory to spectral partitions with volume constraints, providing rigorous blow-up classifications, optimality conditions, and interaction with the domain boundary, with implications for the geometry and regularity of optimal partitions. Overall, the paper advances the understanding of spectral partition problems by integrating penalization techniques, monotonicity formulas, and viscosity methods to achieve sharp free boundary regularity results.

Abstract

This paper is devoted to a complete characterization of the free boundary of all solutions to the following spectral -partition problem with measure and inclusion constraints: where is a bounded domain of , . 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 . 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.
Paper Structure (21 sections, 42 theorems, 280 equations)

This paper contains 21 sections, 42 theorems, 280 equations.

Table of Contents

  1. Introduction
  2. State of the art and related problems
  3. One-phase free boundary problems.
  4. Two-phase free boundary problems.
  5. Strategy of the proof
  6. Notation
  7. A penalized problem. Existence, nondegeneracy, Lipschitz continuity, finite perimeter, density estimates.
  8. Uniform bounds, variational inequalities and regularity of minimizers
  9. Nondegeneracy of eigenfunctions and density estimate from below
  10. The optimal sets have finite perimeter
  11. Two-phase points in the interior and one-phase point on the boundary
  12. Absence of triple points in $\Omega$
  13. Absence of two-phase point on the boundary of $\Omega$
  14. Existence of a one-phase point, shape variations at the free boundary, and equivalence with the constrained problem
  15. Classification of blowups at two-phase points
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^N$ be a bounded Lipschitz domain. Problem eigenvalue_problem admits a solution, and every optimal partition $(\Omega_1,\ldots, \Omega_k)$ of eigenvalue_problem satisfies the following properties. For each $i=1,\ldots,k$, each set $\Omega_i$ is connected, and the boundar Moreover:

Theorems & Definitions (91)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • proof
  • ...and 81 more