An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions. (May 2016)

• Record Type: Journal Article
• Authors: Cabré, Xavier; Serra, Joaquim
• Is Part Of: Nonlinear analysis. Volume 137(2016)
• Page Start: 246
• Page End: 265
• Publication Date: 2016-05
• Subjects: Sums of fractional Laplacians -- One-dimensional symmetry -- Conjecture of De Giorgi; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.12.014
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We study nonlinear elliptic equations for operators corresponding to non-stable Lévy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Lévy processes. We establish the regularity of solutions, as well as sharp energy estimates. As a consequence, we prove a 1-D symmetry result for monotone solutions to Allen–Cahn type equations with a non-stable Lévy diffusion. These operators may still be realized as local operators using a system of PDEs — in the spirit of the extension problem of Caffarelli and Silvestre.