Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius

TL;DR

This paper establishes uniform Lipschitz bounds for eigenfunctions in optimal partition problems with long-range interactions, bridging nonlocal and local cases, and introduces new estimates and monotonicity formulas.

Contribution

It provides the first optimal uniform Lipschitz bounds for eigenfunctions in long-range interaction partition problems, connecting nonlocal and local regimes.

Findings

Established uniform Lipschitz bounds as interaction radius tends to zero

Developed new pointwise eigenfunction estimates and monotonicity formulas

Extended results to singularly perturbed harmonic maps with distance constraints

Abstract

Consider the class of optimal partition problems with long range interactions \[ \inf \left\{ \sum_{i=1}^k \lambda_1(\omega_i):\ (\omega_1,\ldots, \omega_k) \in \mathcal{P}_r(\Omega) \right\}, \] where $\lambda_1(\cdot)$ denotes the first Dirichlet eigenvalue, and $\mathcal{P}_r(\Omega)$ is the set of open $k$-partitions of $\Omega$ whose elements are at distance at least $r$: $\textrm{dist}(\omega_i,\omega_j)\geq r$ for every $i\neq j$. In this paper we prove optimal uniform bounds (as $r\to 0^+$) in $\mathrm{Lip}$-norm for the associated $L^2$-normalized eigenfunctions, connecting in particular the nonlocal case $r>0$ with the local one $r \to 0^+$. The proof uses new pointwise estimates for eigenfunctions, a one-phase Alt-Caffarelli-Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas, combined with elliptic and energy estimates. Our result extends to other contexts, such as singularly perturbed harmonic maps with distance constraints.