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- ItemRegularity and rigidity theorems for a class of anisotropic nonlocal operators(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2016) Farina, Alberto; Valdinoci, EnricoWe 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.