Anisotropic nonlocal operators regularity and rigidity theorems for a class of anisotropic nonlocal operators
- Farina, Alberto
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 35R11 35B53 35R09
- Nonlocal anisotropic integro-differential equations, regularity result
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.