Regularity of the nodal set of segregated critical configurations under a weak reflection law

We deal with a class of Lipschitz vector functions U = (u (1), . . . , u (h) ) whose components are nonnegative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the PohozI dagger aev identity, we prove that the nodal set is a collection of C (1,alpha) hyper-surfaces (for every 0 < alpha < 1), up to a residual set with small Hausdorff dimension. This result applies to the asymptotic limits of reaction-diffusion systems with strong competition interactions, to optimal partition problems involving eigenvalues, as well as to segregated standing waves for Bose-Einstein condensates in multiple hyperfine spin states.