  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,
\[\sum_iu_{i,\beta}\geq c\beta^{-1/4}.\]
