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

Abstract

We deal with a class of Lipschitz vector functions $U=(u_1,...,u_h)$ whose components are non negative, disjointly supported and verify an elliptic equation on each support. Under a weak formulation of a reflection law, related to the Poho\u{z}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.