Uniform Holder bounds and regularity properties of the limiting pro le for highly competing nonlinear systems of Schroedinger equations

We consider systems of k Gross-Pitaevskii equations, arising in the theory of Bose Einstein condensate in multiple species, in the case of large interspecies competitive interactions, both in the focusing and defocusing cases. We prove a priori bounds in the space of Holder continuous functions and, as a consequence, convergence to a limiting space, which can be proved to be Lipschitz continuous. Next we focus on the properties of the interface: we prove regularity and equilibrium at the boundary. The technique is based upon perturbed monotonicity formulas, a little geomatric measure theory and elliptic boundary regularity. As a by product we can prove convergence of both the ground and excited states.