A generalized notion of boundary

Abstract

This paper reports some recent results in the theory of generalized parametric surfaces. The object of this theory, introduced by Young in [18], is to generalize the elementary notions of parametric surface in such a way that a general principle of minimum holds for surface integral variational problems in parametric form. Since nonregular problems are not excluded, the notion of generalized parametric surface is of necessity more general than the usual concepts of surface, involving in many instances both a track (surface in the ordinary sense, possibly mildly discontinuous) and a linear average over a space of Jacobian matrices at each point of the track (for ordinary surfaces the average is the trivial one in which the normal has weight one and all others weight zero) (2). In this paper, we first introduce abstract notions of boundary and track, defined for all non-negative linear functionals L on a certain linear space. Since, in particular, every generalized parametric surface is such a linear functional, these definitions apply to all generalized parametric surfaces. We next specialize to admit only boundaries which correspond in a certain way to boundary curves in the sense of classical analysis, and characterize the set of all L with boundary in this class. Finally, we specialize further by imposing a condition of indecomposibility with respect to the boundary. As an application, some conditions are established for equivalence of indefinite variation problems which have fixed boundary conditions with positive definite problems. 1. Introduction. The main definitions and results of the paper areoutlined in this section. Details and auxiliary results are left to later sections. Let Em (mr> 3) denote the space of all continuous functions f(x, J) of an m-dimensional vector x= [xi, . . ., xm] and an mXm skew-symmetric matrix J = [jr8] of rank 2 which satisfy the homogeneity condition f(x, kJ) = kf(x, J) for k _ O. A mapping x(u, v) from the square U: [O < u < 1; 0 < v <1] to m-space Rm is termed a Dirichlet (parametric) representation if: (1) x(u, v) is absolutely continuous in Tonelli's sense; and