Journal of the European Mathematical Society


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Volume 18, Issue 12, 2016, pp. 2865–2924
DOI: 10.4171/JEMS/656

Published online: 2016-11-21

Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian

Susanna Terracini[1], Gianmaria Verzini[2] and Alessandro Zilio[3]

(1) Università di Torino, Italy
(2) Politecnico di Milano, Italy
(3) Ecole des Hautes Etudes en Sciences Sociales (EHESS), Paris, France

For a class of competition-diffusion nonlinear systems involving the square root of the Laplacian, including the fractional Gross–Pitaevskii system \[ (-\Delta)^{1/2} u_i=\omega_i u_i^3 + \lambda_i u_i - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that $L^\infty$ boundedness implies $\mathcal C^{0,\alpha}$ boundedness for every $\alpha\in[0,1/2)$, uniformly as $\beta\to +\infty$. Moreover we prove that the limiting profile is $\mathcal C^{0,1/2}$.This system arises, for instance, in the relativistic Hartree—Fock approximation theory for $k$-mixtures of Bose–Einstein condensates in different hyperfine states.

Keywords: Square root of the laplacian, spatial segregation, strongly competing systems, optimal regularity of limiting profiles, singular perturbations

Terracini Susanna, Verzini Gianmaria, Zilio Alessandro: Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian. J. Eur. Math. Soc. 18 (2016), 2865-2924. doi: 10.4171/JEMS/656