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The Euler and Weierstrass conditions for nonsmooth variational problems
 A. D. Ioffe,
 R. T. Rockafellar
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Necessary conditions are developed for a general problem in the calculus of variations in which the Lagrangian function, although finite, need not be Lipschitz continuous or convex in the velocity argument. For the first time in such a broadly nonsmooth, nonconvex setting, a full subgradient version of Euler's equation is derived for an arc that furnishes a local minimum in the classical weak sense, and the Weierstrass inequality is shown to accompany it when the arc gives a local minimum in the strong sense. The results are achieved through new techniques in nonsmooth analysis.
This research was supported in part by funds from the U.S.Israel Science Foundation under grant 9000455, and also by the Fund for the Promotion of Research at the Technion under grant 100954 and by the U.S. National Science Foundation under grant DMS9200303.
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 Title
 The Euler and Weierstrass conditions for nonsmooth variational problems
 Journal

Calculus of Variations and Partial Differential Equations
Volume 4, Issue 1 , pp 5987
 Cover Date
 19960101
 DOI
 10.1007/BF01322309
 Print ISSN
 09442669
 Online ISSN
 14320835
 Publisher
 SpringerVerlag
 Additional Links
 Topics
 Keywords

 49K05
 49J52
 58C20
 Authors

 A. D. Ioffe ^{(1)}
 R. T. Rockafellar ^{(2)}
 Author Affiliations

 1. Department of Mathematics, Technion, 34608, Haifa, Israel
 2. Departments of Mathematics and Applied Mathematics, University of Washington, 98195, Seattle, USA