Vol. 15, No. 1, 1965

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ISSN: 0030-8730
Simple areas

Edward Silverman

Vol. 15 (1965), No. 1, 299–303

Let λ 1, E = EN and g be continuous on E × E × E with g(a,,) convex, g(a,kb,kc) = k2g(a,b,c) for all real k and (b2 + c2)λ g(a,b,c) λ(b2 + c2) for all a,b,c E where b2 = b2. If f(a,d e) = minbc=deg(a,b,c) then f is a permissible integrand for the two-dimensional parametric variational problem.

Let γ be a simple closed curve in E, B be the closed unit circle in the plane, C be the collection of functions x continuous on B into E for which x|B γ and D = {x C|x  is a D -map}. Suppose that D is not empty. It was shown in ‘A problem of least area’, [7], that the problem of minimizing I(f) over D is equivalent to minimizing I(g) over D where I(f,x) =f(x,p q),I(g,x) =g(x,p,q),p = xu, q = xv and both integrals are taken over B. The minimizing solution of I(g) is known to have differentiability properties corresponding to g, and this solution also minimizes I(f).

The function f is simple, that is, for each a E, each supporting linear functional to f(a,) is simple. If N = 3, then, of course, each parametric integrand is simple. In this paper we show that for each simple parametric integrand F there exists G, satisfying the conditions imposed upon g, such that F is obtained from G as f was obtained from g.

Mathematical Subject Classification
Primary: 49.00
Received: 13 February 1964
Published: 1 March 1965
Edward Silverman