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

For the singularly perturbed system $Δu_{i,β}=βu_{i,β}\sum_{j\neq i}u_{j,β}^2, \quad 1\leq i\leq N,$ 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, $\sum_iu_{i,β}\geq cβ^{-1/4}.$

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