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)