Generalized Surfaces in the Calculus of Variations. II

Abstract

This note deals with an extension of the idea of surface, which is similar to the writer's generalization' of the notion of curve in the Calculus of Variations. The primary object of such generalizations is to ensure, as far as possible, that every problem of the Calculus of Variations has at least one solution. Their need has been apparent ever since the illuminating remark made by Hilbert:2 Eine jede Aufgabe der Variationsrechnung besitzt eine Losung sobald hinsichtlich der Natur der Grenzbedingungen geeignete einschrankende Annahmen erfillt sind, und, nitigenfalls, der Begriff der Losung eine sinngemdsse Erweiterung erfdhrt. In the existence theorems for variational problems concerning an unknown curve, the customary restriction to regular problems can now be removed by using generalized curves, and applications of these notions have lately been made by McShane.3 Problems of minima of variational problems concerning surfaces are essentially far more complicated than those concerning curves. In this note, however, the additional complication appears only in the proof of certain theorems, not in their statement. The definition of generalized surface given here, and the proof of the existence of a (generalized) solution, are framed for a problem of minimum originally stated in a class I of ordinary surfaces