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

Abstract

For the singularly perturbed system

$$\begin{aligned} \varDelta u_{i,\beta }=\beta u_{i,\beta }\sum _{j\ne i}u_{j,\beta }^2, \quad 1\le i\le N, \end{aligned}$$

we prove that flat segregated interfaces are uniformly Lipschitz as $\beta \rightarrow +\infty$. As a byproduct of the proof we also obtain the optimal lower bound near flat interfaces,

$$\begin{aligned} \sum _iu_{i,\beta }\ge c\beta ^{-1/4}. \end{aligned}$$

.