Some monotonicity results for the fractional Laplacian in unbounded domain. Issue 4 (3rd April 2021)

• Record Type: Journal Article
• Authors: Wu, Leyun; Yu, Mei
• Is Part Of: Complex variables and elliptic equations. Volume 66:Issue 4(2021)
• Page Start: 689
• Page End: 707
• Publication Date: 2021-04-03
• Subjects: The fractional Laplacian -- narrow region principle -- monotonicity -- De Giorgi conjecture; 35S15 -- 35B06 -- 35J60; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2020.1736053
• Languages: English
• ISSNs: 1747-6933

Abstract

Abstract : In this paper, we develop a direct method of moving planes in R n without any decay conditions at infinity for solutions for fractional Laplacian. We first prove a monotonicity result for semi-linear equations involving the fractional Laplacian equation in R n, and we also derive a one-dimensional symmetry result, which indicates that fractional De Giorgi conjecture is valid under some conditions. During these processes, we introduce some new ideas: (i) estimating the singular integrals defining the fractional Laplacian along a sequence of approximate maximum; (ii) analyzing the fractional equations along a sequence of approximate maximum, and then by making translation and taking the limit to derive a limit equation.