Vol. 99, No. 2, 1982

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Isoperimetric eigenvalue problem of even order differential equations

Sui Sun Cheng

Vol. 99 (1982), No. 2, 303–315
Abstract

This paper is concerned with the following eigenvalue problem

{
x(2n) + (− 1)n+1λp(t)x = 0
(2k)         (2k)
x   (0) = 0 = x  (1), k = 0,1,⋅⋅⋅ ,n − 1,
(1)

where p(t) is assumed to be positive and continuous in [0,1]. For the class of functions q(t) which are equimeasurable to p(t), we shall show that the rearrangement of p(t) in symmetrically increasing order maximizes the least positive eigenvalue of (1), while the rearrangement of p(t) in symmetrically decreasing order minimizes it.

Mathematical Subject Classification
Primary: 34B25, 34B25
Secondary: 58F19
Milestones
Received: 7 May 1980
Revised: 13 October 1980
Published: 1 April 1982
Authors
Sui Sun Cheng