%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