Annales Academiĉ Scientiarum Fennicĉ
Mathematica
Volumen 36, 2011, 101-110

A MEAN-VALUE THEOREM FOR SOME EIGENFUNCTIONS OF THE LAPLACE-BELTRAMI OPERATOR ON THE UPPER-HALF SPACE

Sirkka-Liisa Eriksson and Heikki Orelma

Tampere University of Technology, Department of Mathematics
P.O. Box 553, 33101 Tampere, Finland; sirkka-liisa.eriksson 'at' tut.fi

Tampere University of Technology, Department of Mathematics
P.O. Box 553, 33101 Tampere, Finland; heikki.orelma 'at' tut.fi

Abstract. In this paper we study a mean-value property for solutions of the eigenvalue equation of the Laplace-Beltrami operator

\Delta_lbh = -(n - 1)h

with respect to the volume and the surface integrals on the Poincaré upper-half space R₊ⁿ⁺¹ ={(x₀,...,x_n) \in Rⁿ⁺¹ : x_n > 0} with the Riemannian metric ds² = dx₀² + dx₁² + ... + dx_n² / x_n².

2000 Mathematics Subject Classification: Primary 30A05; Secondary 30A05, 30F45.

Key words: Laplace-Beltrami operator, mean-value theorem, hypermonogenic function, hyperbolic harmonic function.

Reference to this article: S.-L. Eriksson and H. Orelma: A mean-value theorem for some eigenfunctions of the Laplace-Beltrami operator on the upper-half space. Ann. Acad. Sci. Fenn. Math. 36 (2011), 101-110.

Full document as PDF file

doi:10.5186/aasfm.2011.3606

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