Vol. 16, No. 1, 1966

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Convexity with respect to Euler-Lagrange differential operators

Junius Colby Kegley

Vol. 16 (1966), No. 1, 87–111

This paper is concerned with the problem of characterizing sub-(L) functions, where L is the Euler-Lagrange operator for the functional Icd[y] = cd[ j=0npj(Djy)2], with n a positive integer, [c,d] a subinterval of a fixed interval [a,b], and p0,p1,,pn continuous real-valued functions on [a,b] with pn(x) > 0 on this interval. Under certain hypotheses on the operator L, it is shown that if f is a function in the domain of L on a subinterval [c,d] of [a,b], then the statement that f is sub-(L) on [c,d] is equivalent to each of the following conditions: (i) (1)nLf(x) 0 on [c,d] (ii) Icd[y] Icd[f] whenever y is a function having continuous derivatives of the first n 1 orders with Dn1y having a piecewise continuous derivative on [c,d] such that Dj1y and Dj1f have the same value at c and at d for j in {1,,n}, and y(x) f(x) 0 on [c,d].

Mathematical Subject Classification
Primary: 34.30
Received: 24 July 1964
Published: 1 January 1966
Junius Colby Kegley