An extension problem for sums of fractional Laplacians and 1-D symmetry of phase transitions

被引:24
作者
Cabre, Xavier [1 ,2 ]
Serra, Joaquim [2 ]
机构
[1] ICREA, Diagonal 647, Barcelona 08028, Spain
[2] Univ Politecn Cataluna, Dept Matemat Aplicada 1, Diagonal 647, E-08028 Barcelona, Spain
关键词
Sums of fractional Laplacians; One-dimensional symmetry; Conjecture of De Giorgi; NONLINEAR EQUATIONS; ELLIPTIC-EQUATIONS; REGULARITY; CONJECTURE; GIORGI;
D O I
10.1016/j.na.2015.12.014
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We study nonlinear elliptic equations for operators corresponding to non-stable Levy diffusions. We include a sum of fractional Laplacians of different orders. Such operators are infinitesimal generators of non-stable (i.e., non self-similar) Levy 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 Levy 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. (c) 2015 Elsevier Ltd. All rights reserved.
引用
收藏
页码:246 / 265
页数:20
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