The Euler and Weierstrass conditions for nonsmooth variational problems

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 90-00455, and also by the Fund for the Promotion of Research at the Technion under grant 100-954 and by the U.S. National Science Foundation under grant DMS-9200303.This article was processed by the author using the style filepljourlm from Springer-Verlag.