On the Variational Principle
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 47, 324-353 (1974) On the Variational Principle I. EKELAND UER Mathe’matiques de la DC&ion, Waiver& Paris IX, 75775 Paris 16, France Submitted by J. L. Lions The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11F’(uJj* < l , i.e., its derivative can be made arbitrarily small. Applications are given to Plateau’s problem, to partial differential equations, to nonlinear eigenvalues, to geodesics on infi- nite-dimensional manifolds, and to control theory. 1. A GENERAL RFNJLT Let V be a complete metric space, the distance of two points u, z, E V being denoted by d(u, v). Let F: I’ -+ 08u {+ co} be a lower semicontinuous function, not identically + 00. In other words, + oo is allowed as a value for F, but at some point w,, , F(Q) is finite. Suppose now F is bounded from below: infF > --co. (l-1) As no compacity assumptions are made, there need be no point where this infimum is attained. But of course, for every E > 0, there is some point u E V such that: infF <F(u) < infF + E.
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