Multiharmonic function

A harmonic function such that the Laplace operator acting on separate groups of independent variables vanishes. More precisely: A function $ u = u ( x _ {1} \dots x _ {n} ) $, $ n \geq 2 $, of class $ C ^ {2} $ in a domain $ D $ of the Euclidean space $ \mathbf R ^ {n} $ is called a multiharmonic function in $ D $ if there exist natural numbers $ n _ {1} \dots n _ {k} $, $ n _ {1} + \dots + n _ {k} = n $, $ n \geq k \geq 2 $, such that the following identities hold throughout $ D $:

$$ \sum _ { \nu = 1 } ^ { {n _ 1 } } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0, $$

$$ \sum _ { \nu = n _ {1} + 1 } ^ { {n _ 1 } + n _ {2} } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0 \dots \sum _ {\nu = n _ {1} + \dots + n _ {k - 1 } + 1 } ^ { n } \frac{\partial ^ {2} u }{\partial x _ \nu ^ {2} } = 0. $$

An important proper subclass of the class of multiharmonic functions consists of the pluriharmonic functions (cf. Pluriharmonic function), for which $ n = 2m $, $ n _ {j} = 2 $, $ j = 1 \dots m $, i.e. $ k = m $, and which also satisfy certain additional conditions.

References

[1]	E.M. Stein, G. Weiss, "Introduction to harmonic analysis on Euclidean spaces" , Princeton Univ. Press (1971)

Comments

Multiharmonic functions are also called multiply harmonic functions. It was shown by P. Lelong that if $ u $ is separately harmonic, that is, $ u $ is harmonic as a function of $ x _ {n _ {j} + 1 } \dots x _ {n _ {j+} 1 } $( $ j= 0 \dots k $; $ n _ {0} = 0 $) while the other variables remain fixed, then $ u $ is multiharmonic. A different proof is due to J. Siciak. See [a1].

References

[a1]	M. Hervé, "Analytic and plurisubharmonic functions" , Lect. notes in math. , 198 , Springer (1971)