Eigenfunctions for quasi-Laplacian. (October 2019)

• Record Type: Journal Article
• Authors: Chen, Min; Li, Jiayu
• Is Part Of: Nonlinear analysis. Volume 187(2019)
• Page Start: 205
• Page End: 213
• Publication Date: 2019-10
• Subjects: 47F05 -- 58C40; Quasi-Laplacian -- Singularity -- Eigenfunction; 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.2019.04.013
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: To study the regularity of heat flow, Lin and Wang (1999) introduced the quasi-harmonic sphere, which is a harmonic map from M = ( R m, e − | x | 2 2 ( m − 2 ) d s 0 2 ) to N with finite energy. Here d s 0 2 is Euclidean metric in R m . Ding and Yongqiang (2006) showed that if the target is a sphere, any equivariant quasi-harmonic spheres is discontinuous at infinity. The metric g = e − | x | 2 2 ( m − 2 ) d s 0 2 is quite singular at infinity and it is not complete. In this paper, we mainly study the eigenfunction of Quasi-Laplacian Δ g = e | x | 2 2 ( m − 2 ) ( Δ g 0 − ∇ g 0 h ⋅ ∇ g 0 ) = e | x | 2 2 ( m − 2 ) Δ h for h = | x | 2 4 . In particular, we show that there are infinite number of eigenvalues (of the quasi-Laplacian Δ g ) of which the corresponding eigenfunctions are discontinuous at ∞ and any non-constant eigenfunction of drifted Laplacian Δ h = Δ g 0 − ∇ g 0 h ⋅ ∇ g 0 is discontinuous at infinity.