Algebraic structure of quasiradial solutions to the γ-harmonic equation

Vladimir G. Tkachev

Vol. 226 (2006), No. 1, 179–200

DOI: 10.2140/pjm.2006.226.179

Abstract

We obtain an explicit representation for quasiradial γ-harmonic functions, which shows that these functions have an essentially algebraic nature. We give a complete description of all γ that admit algebraic quasiradial solutions. Unlike the cases γ = ∞ and γ = 1, only finitely many algebraic solutions are shown to exist for any fixed |γ| > 1. A special extremal series of γ corresponds exactly to the well known ideal m-atomic gas adiabatic constant γ = (2m + 3)∕(2m + 1).

Keywords

γ-harmonic function, quasiradial solution, algebraic function

Mathematical Subject Classification 2000

Primary: 35J15

Secondary: 31A05, 14H05

Milestones

Received: 16 October 2004

Revised: 28 January 2005

Accepted: 15 February 2005

Published: 1 July 2006