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 $$\begin{aligned} \inf \left\{ \sum _{i=1}^k \lambda _1(\omega _i):\ (\omega _1,\ldots , \omega _k) \in \mathcal {P}_r(\Omega ) \right\} , \end{aligned}$$ 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: $${{\,\mathrm{dist}\,}}(\omega _i,\omega _j)\ge r$$ for every $$i\ne j$$ . In this paper we prove optimal uniform bounds (as $$r\rightarrow 0^+$$ ) in $$\mathrm {Lip}$$ –norm for the associated $$L^2$$ -normalized eigenfunctions, connecting in particular the nonlocal case $$r>0$$ with the local one $$r \rightarrow 0^+$$ . 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.