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) |
Multiharmonic function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Multiharmonic_function&oldid=47925