Proceedings of Symposia in Pure Mathematics

Volume 57 (1995)

An Improvement of the Osserman Constant

for the Bass Note of a Drum

RODRIGO BANUELOS AND TOM CARROLL

0. Introduction

The following result of Hayman [9] answered an important question in the

study of vibrating membranes raised by Polya and Szego in [15, p. 16]:

THEOREM

(Hayman). Let D be a simply connected domain in the plane.

Let RD be the radius of the largest disc contained in D, and let XD be the

first Dirichlet eigenvalue for the Laplacian in D. There is a universal constant

a such that

(0.1)

^DalR2D.

Thus for a drum to produce an arbitrarily low tone it must necessarily

contain an arbitrarily large circular drum. The domain monotonicity of the

eigenvalue, which follows either from the variational formula or the proba-

bilistic characterization of kD (see below), implies that

(0.2)

iDhlR2D,

where j0 is the smallest positive zero of the first Bessel function. The in-

equality (0.2) is sharp with equality attained when D is a disc. The Hayman

inequality was of great interest to analysts, geometers, and probabilists when

it appeared in 1976, and there have been many efforts to find the best con-

stant a and to identify the extremal domain. Hayman's original proof gave

a = 1/900. The references [1, 4, 5, 8, 14, 16], contain various proofs and

extensions of Hayman's theorem with various values of a . In [14] Osserman

proves the inequality with a = 1/4 and suggests that this may be a candidate

for the best constant. In Banuelos and Carroll [2], we prove that a 0.6197 .

In this note we present a different and simpler proof that a 0.325 , which

1991 Mathematics Subject Classification. Primary 31A35, 60J65.

The first author was supported in part by National Science Foundation grant 8958449-DMS.

©1995 American Mathematical Society

0082-0717/95 $1.00+ $.25 per page

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http://dx.doi.org/10.1090/pspum/057/1335458