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

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 ${\lambda }_1(\cdot )$ 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 $r$: $\text{dist}({\omega }_i,{\omega }_j)\ge r$ for every $i\ne j$. 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.

23 pages