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

TL;DR

This paper proves that under a weak reflection law, the nodal set of certain segregated elliptic systems forms smooth hyper-surfaces, with implications for reaction-diffusion systems, eigenvalue partitions, and Bose-Einstein condensates.

Contribution

It establishes regularity of the nodal set for segregated solutions under a weak reflection law, extending understanding of their geometric structure.

Findings

Nodal set consists of $C^{1,eta}$ hyper-surfaces.

Residual set with small Hausdorff dimension where regularity may fail.

Applicable to reaction-diffusion limits, eigenvalue partitions, and Bose-Einstein condensates.

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.