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Estimates on the Green's function of second-order elliptic operators in ℝN

Published online by Cambridge University Press:  14 November 2011

Adrian T. Hill
Adrian T. Hill
Affiliation:
School of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, U.K., E-mail: ath@maths.bath.ac.uk
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Abstract

Sharp upper and lower pointwise bounds are obtained for the Green's function of the equation

for λ> 0. Initially, in a Cartesian frame, it is assumed that . Pointwise estimates for the heat kernel of ut + Lu = 0, recently obtained under this assumption, are Laplace-transformed to yield corresponding elliptic results. In a second approach, the coordinate-free constraint is imposed. Within this class of operators, the equations defining the maximal and minimal Green's functions are found to be simple ODEs when written in polar coordinates, and these are soluble in terms of the singular Kummer confluent hypergeometric function. For both approaches, bounds on are derived as a consequence.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 128 , Issue 5 , 1998 , pp. 1033 - 1051
Copyright
Copyright © Royal Society of Edinburgh 1998

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