For a class of systems of semi-linear elliptic equations, including (Formula presented.), for p = 2 (variational-type interaction) or p = 1 (symmetric-type interaction), we prove that uniform L^∞ boundedness of the solutions implies uniform boundedness of their Lipschitz norm as β→+∞, lthat is, in the limit of strong competition. This extends known quasi-optimal regularity results and covers the optimal case for this class of problems. The proofs rest on monotonicity formulae of Alt–Caffarelli–Friedman and Almgren type in the variational setting, and on the Caffarelli–Jerison–Kenig almost monotonicity formula in the symmetric one

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross-Pitaevskii systemwe prove that L ∞ boundedness implies C 0,α boundedness for every α ∈ [0, 1/2), uniformly as β → +∞. Moreover we prove that the limiting profile is C 0,1/2 . This system arises, for instance, in the relativistic Hartree-Fock approximation theory for k-mixtures of Bose-Einstein condensates in different hyperfine states. thus focusing on the singular limit problem obtained when the (positive) parameter β, accounting for the competitive interactions, diverges to ∞. Among the others, the cases f i (s) = r i s(1−s/K i ), g ij (s) = a ij s (logistic internal dynamics with Lotka-Volterra competition) and f i (s) = ω i s 3 + λ i s, g ij (s) = a ij s 2 (focusing-defocusing 2010 Mathematics Subject Classification. Primary: 35J65; secondary: 35B40 35B44 35R11 81Q05 82B10.

For a class of competition-diffusion nonlinear systems involving the s-power of the Laplacian, s ∈ (0, 1), of the formwe prove that L ∞ boundedness implies C 0,α boundedness for α > 0 sufficiently small, uniformly as β → +∞. This extends to the case s = 1/2 part of the results obtained by the authors in the previous paper [arXiv:1211.6087v1].2010 Mathematics Subject Classification. Primary: 35J65; secondary: 35B40 35B44 35R11 81Q05 82B10.

Abstract. We study interior regularity issues for systems of elliptic equations of the typeThe paper is devoted to the derivation of C 0,α estimates that are uniform in the competition parameter β > 0, as well as to the regularity of the limiting free-boundary problem obtained for β → +∞.The main novelty of the problem under consideration resides in the non-trivial grouping of the densities: in particular, we assume that the interaction parameters a ij are only non-negative, and thus may vanish for specific couples (i, j). As a main consequence, in the limit β → +∞, densities do not segregate pairwise in general, but are grouped in classes which, in turn, form a mutually disjoint partition. Moreover, with respect to the literature, we consider more general forcing terms, sign-changing solutions, and an arbitrary p > 0. In addition, we present a regularity theory of the emerging free-boundary, defined by the interface among different segregated groups.These equations are very common in the study of Bose-Einstein condensates and are of key importance for the analysis of optimal partition problems related to high order eigenvalues.

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