Inequalities for harmonic functions on spheroids and their applications


Zahariuta V.

INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol.50, no.2, pp.1047-1075, 2001 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 50 Issue: 2
  • Publication Date: 2001
  • Title of Journal : INDIANA UNIVERSITY MATHEMATICS JOURNAL
  • Page Numbers: pp.1047-1075

Abstract

Hadamard-type interpolational inequalities for norms of harmonic functions are studied for confocal prolate and oblate spheroids. It is shown that the optimal level domains in such inequalities may be non-spheroidal. Moreover, in contrary with the case of analytic functions, there is an unremovable gap between the corresponding optimal level domains for inner and outer versions of Hadamard-type inequalities for harmonic functions. These results are based on some special asymptotical formulas for associated Legendre functions P-n(m), Q(n)(m), when the ratio m/n tends to some number gamma is an element of [0, 1]. The case of spheroids is studied in the frame of the general theory of so-called Lh-functions and extremal Lh-functions (Lh-potentials) (developed earlier by the author), which serves the harmonic functions case much as the theory of plurisubharmonic and maximal plurisubharmonic functions (pluripotentials) (Lelong, Bremermann, Siciak, Zahariuta, Bedford-Taylor, Sadullayev et al.) does in the case of analytic functions of several variables.