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

Consider the class of optimal partition problems with long range interactionsinf {Sigma(k)(i=1) lambda(1)(omega(i)) : (omega(1), . . . , omega(k)) is an element of P-r (Omega)},where lambda(1)(.) denotes the first Dirichlet eigenvalue, and P-r (Omega) is the set of open k-partitions of Omega whose elements are at distance at least r: dist(omega(i), omega(j)) >= r for every i not equal j. In this paper we prove optimal uniform bounds (as r -> 0(+)) in Lip-norm for the associated L-2-normalized eigenfunctions, connecting in particular the nonlocal case r > 0 with the local one r -> 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.