For the positive solutions of the Gross-Pitaevskii system we prove that L∞-boundedness implies C0,α-boundedness for every α∈(0,1), uniformly as β→+∞. Moreover, we prove that the limiting profile as β→+∞ is Lipschitz-continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given
Noris, B., Terracini, S., Tavares, H., Verzini, G. (2010). Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition. COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, 63(3), 267-302 [10.1002/cpa.20309].
Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition
Noris, B
;Terracini, S
;
2010
Abstract
For the positive solutions of the Gross-Pitaevskii system we prove that L∞-boundedness implies C0,α-boundedness for every α∈(0,1), uniformly as β→+∞. Moreover, we prove that the limiting profile as β→+∞ is Lipschitz-continuous. The proof relies upon the blowup technique and the monotonicity formulae by Almgren and Alt, Caffarelli, and Friedman. This system arises in the Hartree-Fock approximation theory for binary mixtures of Bose-Einstein condensates in different hyperfine states. Extensions to systems with k > 2 densities are given
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