Metric Methods in Calculus of Variations

Abstract

1. In each problem of calculus of variations there are four fundamental data: (I) A point-space S or a domain of a point-space. (II) A function φ from which there is derived a functional in a class P of entities of S whose dimensions are ≧ 1. With each element C of & there is associated a number, say λ_>φ,(C), by a process of integration of ϕ along C. The class P; is a topological space in which a metric may be introduced. (III) A sub-class G of G, the class of admissible entities. (IV) A sub-class P ^*₀ of ?*. A main problem is to find conditions under which P ^*₀ contains elements minimizing λ_>ϕ(C) or stationary for λϕ(C) with respect to all elements of S*. The solution of this problem is the more general the fewer restrictions are imposed on S, ϕ and S*, and the more restrictions are imposed on S*. In fact, one of the main tendencies in calculus of variations has been to find solutions of this problem more and more general in the mentioned sense.¹

Since the generality of a solution depends, so to speak, on four parameters S, ϕ, G, G, it is obvious that not all solutions may be arranged into a linear order of increasing generality.