Mean Value Properties for the Weinstein Equation Using the Hyperbolic Metric

作者：Eriksson S L^*; Orelma H

来源：Complex Analysis and Operator Theory, 2013, 7(5): 1609-1621.

DOI：10.1007/s11785-012-0280-4

• Hyperbolic metric
• Laplace-Beltrami
• Weinstein equation
• Harmonic
• Conformal manifolds

In this paper we consider solutions of the Weinstein equation %26lt;br%26gt;Delta u - k/chi(n) partial derivative u/partial derivative chi(n) + l/chi(2)(n)u = 0 %26lt;br%26gt;on some open subset Omega subset of R-n boolean AND {X-n %26gt; 0} subject to the conditions 4l %26lt;= (k + 1)(2). If l = 0, the operator X-n(2k/n-2) (Delta u - k/chi(n) partial derivative u/partial derivative chi(n)) is the Laplace-Beltrami operator with respect to the Riemannian metric ds(2) = X-n(-2k/n-2) (Sigma(n)(i=1) dx(i)(2)) In case k = n - 2 the Riemannian metric is the hyperbolic distance of Poincar, upper half space. The Weinstein equation is connected to the axially symmetric potentials. The solutions of of the Weinstein equation form a so-called Brelot harmonic space and therefore it is known they satisfy the mean value properties with respect to the harmonic measure. We present the explicit mean value properties which give a formula for a harmonic measure evaluated in the center point of the hyperbolic ball. The key idea is to transform the solutions to the eigenfunctions of the Laplace-Beltrami operator in the Poincar, upper half-space model.

• 出版日期2013-10