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arXiv · 2106.03661

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

Nicola Soave·Hugo Tavares·Alessandro Zilio
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Abstract

Consider the class of optimal partition problems with long range interactions \[ \inf \left\{ \sum_{i=1}^k λ_1(ω_i):\ (ω_1,\ldots, ω_k) \in \mathcal{P}_r(Ω) \right\}, \] where $λ_1(\cdot)$ denotes the first Dirichlet eigenvalue, and $\mathcal{P}_r(Ω)$ is the set of open $k$-partitions of $Ω$ whose elements are at distance at least $r$: $\textrm{dist}(ω_i,ω_j)\geq r$ 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.

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Nicola Soave, Hugo Tavares, Alessandro Zilio. 2021-06-07. Free boundary problems with long-range interactions: uniform Lipschitz estimates in the radius. https://arxiv.org/abs/2106.03661

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