Calculus of Variations and Geometric Measure Theory
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A. Farina - E. Valdinoci

REGULARITY AND RIGIDITY THEOREMS FOR A CLASS OF ANISOTROPIC NONLOCAL OPERATORS

created by farina on 15 Jan 2016

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Submitted Paper

Inserted: 15 jan 2016
Last Updated: 15 jan 2016

Year: 2016

Abstract:

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.


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