%0 Journal Article
%T Regularity and rigidity theorems for a class of anisotropic nonlocal operators
%+ Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA)
%A Farina, Alberto
%A Valdinoci, Enrico
%< avec comité de lecture
%@ 0025-2611
%J Manuscripta mathematica
%I Springer Verlag
%V 153
%N 1-2
%P 53-70
%8 2017
%D 2017
%Z 1512.06509
%R 10.1007/s00229-016-0875-6
%Z Mathematics [math]Journal articles
%X We consider here operators which are sum of (possibly) fractional derivatives, with (possibly different) order. The main constructive assumption is that the operator is of order 2 in one variable. By constructing an explicit barrier, we prove a Lipschitz estimate which controls the oscillation of the solutions in such direction with respect to the oscillation of the nonlinearity in the same direction. As a consequence, we obtain a rigidity result that, roughly speaking, states that if the nonlinearity is independent of a coordinate direction, then so is any global solution (provided that the solution does not grow too much at infinity). A Liouville type result then follows as a byproduct.
%G English
%L hal-03621418
%U https://u-picardie.hal.science/hal-03621418
%~ CNRS
%~ UNIV-PICARDIE
%~ INSMI
%~ U-PICARDIE
%~ LAMFA