Eigenfunctions for quasi-Laplacian. (October 2019) - Chen, Min; Li, Jiayu
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.