Integral representation and asymptotic behavior of harmonic functions in half space

Abstract

Using Carleman's formula of a harmonic function in the half space and Nevanlinna's representation of a harmonic function in the half sphere, we prove that a harmonic function, whose positive part satisfies a slowly growing condition, can be represented by a certain integral. This improves some classical Poisson integrals for harmonic functions.

Asymptotic behavior of fractional Laplacians in the half space

“…One can find interesting properties and interpretations of fractional calculus in [14,15,17], which also give a useful mathematical tool for modeling many process in nature. Moreover, this paper provides the details of the asymptotic properties for the fractional Laplacian operator, as the applications of theory, i.e., Theorem 3.1 is the generation of asymptotic behaviors in [24][25][26], especially when a ¼ 2, Theorem 3.1 reduces to the first result of paper [25]; when a ¼ 2; m ¼ 1, Theorem 3.3 reduces to the second result of paper [27].…”

Phragmén-Lindelöf theorems of subharmonic functions and their applications in the Half space

“…文献 [16] 利用调和函数的 Carlmen 公式 (1.2), 证明了若半空间中一个调和函数的正部满足一个缓 慢增长条件, 则该函数可以由其边界超平面上的积分表示出来. 受到文献 [2,3,15,16] [3,14] , Levi [2] [15] , [3,14] 和 Levi [2] 的结果.…”