HAL CCSD
Regularity and rigidity theorems for a class of anisotropic nonlocal operators
Farina, Alberto
Valdinoci, Enrico
Laboratoire Amiénois de Mathématique Fondamentale et Appliquée - UMR CNRS 7352 (LAMFA) ; Université de Picardie Jules Verne (UPJV)-Centre National de la Recherche Scientifique (CNRS)
International audience
ISSN: 0025-2611
EISSN: 1432-1785
Manuscripta mathematica
Springer Verlag
hal-03621418
https://u-picardie.hal.science/hal-03621418
https://u-picardie.hal.science/hal-03621418
Manuscripta mathematica, 2017, 153 (1-2), pp.53-70. ⟨10.1007/s00229-016-0875-6⟩
ARXIV: 1512.06509
info:eu-repo/semantics/altIdentifier/arxiv/1512.06509
DOI: 10.1007/s00229-016-0875-6
info:eu-repo/semantics/altIdentifier/doi/10.1007/s00229-016-0875-6
en
[MATH]Mathematics [math]
info:eu-repo/semantics/article
Journal articles
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.
2017