# Regularity and rigidity theorems for a class of anisotropic nonlocal operators

Alberto Farina, Enrico Valdinoci

Research output: Contribution to journalArticle

9 Citations (Scopus)

### 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.

Original language English 53-70 18 Manuscripta Mathematica 153 1-2 https://doi.org/10.1007/s00229-016-0875-6 Published - 1 May 2017

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Rigidity
Regularity
Operator
Theorem
Nonlinearity
Oscillation
Fractional Derivative
Global Solution
Lipschitz
Infinity
Estimate
Class

### Cite this

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In: Manuscripta Mathematica, Vol. 153, No. 1-2, 01.05.2017, p. 53-70.

Research output: Contribution to journalArticle

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T1 - Regularity and rigidity theorems for a class of anisotropic nonlocal operators

AU - Farina, Alberto

AU - Valdinoci, Enrico

PY - 2017/5/1

Y1 - 2017/5/1

N2 - 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.

AB - 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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