Semicontinuity of integrals

Abstract

where A is a space of any dimension, ,u a general measure function on a a-algebra of subsets of A, where p(w): A -+ E, o(w): A -+ F are any two maps, and f any scalar function defined on E x F. The semicontinuity is with respect to an appropriate convergence on (p, a, iu). The second map a: A -+ F may be interpreted as a very general substitute for the usual normal vector. Both parametric and nonparametric integrals can be interpreted as particular cases of (*). A general theorem of this nature was proposed to the writer by Professor L. Cesari. He suggested it as a natural development of his treatment of integrals over a general variety [3], [4], abstracting and unifying the work of Weierstrass, Tonelli, and others on curve integrals, and of Cesari himself on surface integrals. In particular, Cesari has shown [4] that integrals of this nature can be expressed quite generally as measure integrals with which we are concerned in this paper. The actual form of our general semicontinuity theorem was influenced particularly by the special cases treated in [5], [6], [9], [10], and [12]. In later sections, we apply our general theorem to special cases: parametric curve integrals [8], [10], parametric curve integrals involving higher derivatives [5], [6], parametric surface integrals [1], [12], nonparametric curve integrals [8], [9], [10], and nonparametric integrals [8]. In each case, we verify general conditions assumed in our semicontinuity theorem to obtain semicontinuity theorems in the corresponding section of the calculus of variations. Known semicontinuity theorems of Tonelli, Cesari, Cinquini, and Turner, each involving a particular topology, are so obtained as particular cases of only one general statement concerning integral (*).