Simple areas

Edward Silverman

Vol. 15 (1965), No. 1, 299–303

DOI: 10.2140/pjm.1965.15.299

Abstract

Let $\lambda \geqq 1$, $E=E^N$ and $g$ be continuous on $E\times E\times E$ with $g\left(a,\cdot ,\cdot \right)$ convex, $g\left(a,kb,kc\right)=k^{2}g\left(a,b,c\right)$ for all real $k$ and $\left(b^2+c^2\right)∕\lambda \leqq g\left(a,b,c\right)\leqq \lambda \left(b^2+c^2\right)$ for all $a,b,c\in E$ where $b^2=\parallel b{\parallel }^2$. If $f\left(a,d\wedge e\right)=\underset{b\wedge c=d\wedge e}{min}g\left(a,b,c\right)$ then $f$ is a permissible integrand for the two-dimensional parametric variational problem.

Let $\gamma$ 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 $x|\partial B\in \gamma$ and $D=\left\{x\in C|x\text{ is a }D\text{ -map}\right\}$. Suppose that $D$ is not empty. It was shown in ‘A problem of least area’, [7], that the problem of minimizing $I\left(f\right)$ over $D$ is equivalent to minimizing $I\left(g\right)$ over $D$ where $I\left(f,x\right)=\iint f\left(x,p\wedge q\right),I\left(g,x\right)=\iint g\left(x,p,q\right),p=x_u$, $q=x_v$ and both integrals are taken over $B$. The minimizing solution of $I\left(g\right)$ is known to have differentiability properties corresponding to $g$, and this solution also minimizes $I\left(f\right)$.

The function $f$ is simple, that is, for each $a\in E$, each supporting linear functional to $f\left(a,\cdot \right)$ 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

Milestones

Received: 13 February 1964

Published: 1 March 1965