Selecta Mathematica pp 391-397 | Cite as

# Metric Methods in Calculus of Variations

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## 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}

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