Integral representations of harmonic functions in half spaces

作者：Guantie Deng^*

来源：Bulletin des Sciences Mathematiques, 2007, 131(1): 53-59.

DOI：10.1016/j.bulsci.2006.03.008

• harmonic function
• integral representation
• half-space

In this paper, using a modified Poisson kernel in an upper half-space, we prove that a harmonic function u(z) in a upper half space with its positive part u(+)(x) = max{u(x), 0} satisfying a slowly growing condition can be represented by its integral in the boundary of the upper half space, the integral representation is unique up to the addition of a harmonic polynomial, vanishing in the boundary of the upper half space and that its negative part u(-)(x) = max(-u(x), 0) can be dominated by a similar slowly growing condition, this improves some classical result about harmonic functions in the upper half space.

• 单位
  北京师范大学