Vol. 30, No. 2, 1969

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Best constants in a class of integral inequalities

David W. Boyd

Vol. 30 (1969), No. 2, 367–383
Abstract

In this paper a method is developed for determining best constants in inequalities of the following form:

∫ b  p (n)q           ∫ b  (n)r       (p+q)∕r
a |y| |y  | w(x)dx ≦ K { a |y |m (x)dx }     ,

where y(a) = y(a) = = y(n1)(a) = 0 and y(n1) is absolutely continuous.

It is first shown that for a certain class of m and w, equality can be attained in the inequality. Applying variational techniques reduces the determination of the best constant to a nonlinear eigenvalue problem for an integral operator. If m and w are sufficiently smooth this reduces further to a boundary value problem for a differential equation. The method is illustrated by determinin g the best constants in case (a,b) is a finite interval, m(x) w(x) 1, and n = 1.

Mathematical Subject Classification
Primary: 26.70
Milestones
Received: 27 September 1968
Published: 1 August 1969
Authors
David W. Boyd
Department of Mathematics
University of British Columbia
Vancouver BC V6T 1Z2
Canada
http://www.math.ubc.ca/~boyd/boyd.html