Uniform Lipschitz regularity of flat segregated interfaces in a singularly perturbed problem

TL;DR

This paper proves that flat interfaces in a singularly perturbed system are uniformly Lipschitz continuous and establishes an optimal lower bound near these interfaces, advancing understanding of the system's regularity properties.

Contribution

It demonstrates uniform Lipschitz regularity of flat interfaces and provides an optimal lower bound near interfaces in a singularly perturbed system.

Findings

Flat interfaces are uniformly Lipschitz.

Established the optimal lower bound near interfaces.

Enhanced understanding of regularity in singular perturbation problems.

Abstract

For the singularly perturbed system \[\Delta u_{i,\beta}=\beta u_{i,\beta}\sum_{j\neq i}u_{j,\beta}^2, \quad 1\leq i\leq N,\] we prove that flat interfaces are uniformly Lipschitz. As a byproduct of the proof we also obtain the optimal lower bound near the flat interfaces, \[\sum_iu_{i,\beta}\geq c\beta^{-1/4}.\]