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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 thefirst 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 weprove optimal uniform bounds (as $r\to 0^+$) in $\mathrm{Lip}$-norm for theassociated $L^2$-normalized eigenfunctions, connecting in particular thenonlocal case $r>0$ with the local one $r \to 0^+$.  The proof uses new pointwise estimates for eigenfunctions, a one-phaseAlt-Caffarelli-Friedman and the Caffarelli-Jerison-Kenig monotonicity formulas,combined with elliptic and energy estimates. Our result extends to othercontexts, such as singularly perturbed harmonic maps with distance constraints.

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