On metaharmonic functions. (English) Zbl 1290.31009

Vekua, Ilia

Lect. Notes TICMI 14, 62 p. (2013).

This work (of the late I. N. Vekua (23.IV.1907 – 2.XII.1977) translated from the Russian by Ts. Gabeskiria) concerns the study of the differential equation \[ \Delta^n u+a_1\Delta^{n-1} u+\dots+a_n u=0\qquad\text{in}\qquad T\subset {\mathbb R}^p, (*) \] where \(T\) is a domain, \(n\geq 1\), \(a_1,a_2,\dots,a_n\) are complex numbers. Solutions \(u:T\to {\mathbb C}\) of equation \((*)\) are called \(n\)-metaharmonic functions. The first half of this work is concerned with the case \(n=1\). Here Green’s formulae are derived together with various integral representations both in finite and infinite domains. Further, the author describes the series expansion of 1-metaharmonic functions in terms of Hankel and hyperspherical functions.
The second half of this work deals with the case \(n>1\) where some representation formulae are obtained for \(n\)-metaharmonic functions. These representation formulae are next used to discuss the Riquier’s boundary value problem.
For the original text see Tr. Tbilis. Mat. Inst. 12, 105–174 (1943; Zbl 0063.07996). See also I. N. Vekua. Complex representation of the general solution of the equations of the steady two dimensional problem of the theory of elasticity, Dokl. Akad. Nauk SSSR 16, No. 3, 163–168 (1937); Soobshch. Akad. Nauk Gruz. SSR 2, 29–34 (1941; Zbl 0063.07995); Soobshch. Akad. Nauk Gruz. SSR 3, 307-314 (1942; Zbl 0063.08016); Soobshch. Akad. Nauk Gruz. SSR 4, 281–288 (1943; Zbl 0063.08021).

Reviewer: Marius Ghergu (Dublin)

MSC:

31B99	Higher-dimensional potential theory
31C99	Generalizations of potential theory
01A75	Collected or selected works; reprintings or translations of classics

Keywords:

\(n\)-metaharmonic functions; Hankel functions; hyperspherical functions; Vekua-Rellich lemma

Citations:

Zbl 0063.07996; Zbl 0063.07995; Zbl 0063.08016; Zbl 0063.08021