https://u-picardie.hal.science/hal-03621418Farina, AlbertoAlbertoFarinaLAMFA - Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 - UPJV - Université de Picardie Jules Verne - CNRS - Centre National de la Recherche ScientifiqueValdinoci, EnricoEnricoValdinociRegularity and rigidity theorems for a class of anisotropic nonlocal operatorsHAL CCSD2017[MATH] Mathematics [math]DESSAIVRE, Louise2022-03-28 10:57:072023-03-24 14:53:262022-03-28 10:57:07enJournal articles10.1007/s00229-016-0875-61We 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.