Metric Methods in Calculus of Variations
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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 P0* of ?*. A main problem is to find conditions under which P0* 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
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