Sublinear functions of measures and variational integrals

Abstract

The original purpose of this paper was to establish that certain non-parametric variational integrals may be considered as measures on the domain of definition of the admissible functions. This was accomplished by exhibiting an explicit formula for the functional involved, the formula containing within itself, in the case of the surface area integral, a proof both of Tonelli's celebrated theorem on Lebesgue area and of a less well known but deeper result of Verchenko. E . This set function is in some ways analogous to the total variation of ;

Lower semicontinuity of surface energies

“…The following lower semicontinuity result was proved independently by GOFFMAN & SERRIN [21] and by RESHETNYAK [30].…”

“…For the remaining of this section, we refer the reader to GOFFMAN & SERRIN [21]. Let p be a IRP -valued measure in Cl with polar decomposition dp = a dX, and let f e Co(£ixlR p ).…”

A note on the derivation of rigid-plastic models

“…while, for any p ∈ M(Ω ∪ Γ D ; M n×n D ), the dissipation energy is the convex functional of measure (see [9,6])…”

Derivative Measures

“…The measure a, may also be obtained as a lower semicontinuous extension of an area functional to the set of summable functions from the set of piecewise linear functions, and then by extending this area from a function on rectangles to a measure on Borel sets. A general discussion of numerical valued measures of "area" type associated with vector valued measures may be found in [3] where further references are given. Lemma 1.…”

“…,vn be the directional derivatives of / in these coordinate directions. By Krickeberg's lemma (see [3] for a proof), Vj(S) = fR»-\ Vj(S, x)dx, where x varies over the space R"~l determined by the n -1 coordinates left after the /th coordinate in this system is omitted, and vj(S, x) is the variation measure of / in the /th coordinate on the intersection of S with the line obtained by fixing the other coordinates at x. By our choice of coordinates, vJ(S, x) = 0 almost everywhere, so that v¡(S) = 0, / = 1, .…”