### 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 |
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Pages (from-to) | 53-70 |

Number of pages | 18 |

Journal | Manuscripta Mathematica |

Volume | 153 |

Issue number | 1-2 |

DOIs | |

Publication status | Published - 1 May 2017 |

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### Cite this

*Manuscripta Mathematica*,

*153*(1-2), 53-70. https://doi.org/10.1007/s00229-016-0875-6

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*Manuscripta Mathematica*, vol. 153, no. 1-2, pp. 53-70. https://doi.org/10.1007/s00229-016-0875-6

**Regularity and rigidity theorems for a class of anisotropic nonlocal operators.** / Farina, Alberto; Valdinoci, Enrico.

Research output: Contribution to journal › Article

TY - JOUR

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.

KW - 35B53

KW - 35R09

KW - 35R11

UR - http://www.scopus.com/inward/record.url?scp=84982170185&partnerID=8YFLogxK

U2 - 10.1007/s00229-016-0875-6

DO - 10.1007/s00229-016-0875-6

M3 - Article

VL - 153

SP - 53

EP - 70

JO - Manuscripta Mathematica

JF - Manuscripta Mathematica

SN - 0025-2611

IS - 1-2

ER -