Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

Susanna Terracini; Gianmaria Verzini; Alessandro Zilio

doi:10.3934/dcds.2014.34.2669

Article Contents

2014, Volume 34, Issue 6: 2669-2691. Doi: 10.3934/dcds.2014.34.2669

This issue Previous Article On the multiplicity of nonnegative solutions with a nontrivial nodal set for elliptic equations on symmetric domains Next Article

Uniform Hölder regularity with small exponent in competition-fractional diffusion systems

1.
Dipartimento di Matematica "Giuseppe Peano", Università degli Studi di Torino, Via Carlo Alberto 10, 10123 Torino
2.
Dipartimento di Matematica "Francesco Brioschi", Politecnico di Milano, p.za Leonardo da Vinci 32, 20133 Milano

Received: February 28, 2013

Revised: May 31, 2013

Published: May 2014

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Abstract

For a class of competition-diffusion nonlinear systems involving the $s$-power of the laplacian, $s\in(0,1)$, of the form \[ (-\Delta)^{s} u_i=f_i(u_i) - \beta u_i\sum_{j\neq i}a_{ij}u_j^2,\qquad i=1,\dots,k, \] we prove that $L^\infty$ boundedness implies $\mathcal{C}^{0,\alpha}$ boundedness for $\alpha>0$ sufficiently small, uniformly as $\beta\to +\infty$. This extends to the case $s\neq1/2$ part of the results obtained by the authors in the previous paper [arXiv: 1211.6087v1].

Keywords:

Fractional laplacian,
spatial segregation,
strongly competing systems,
optimal regularity of limiting profiles,
singular perturbations.

Mathematics Subject Classification: Primary: 35J65; Secondary: 35B40, 35B44, 35R11, 81Q05, 82B10.

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