Calculus of Variations and Partial Differential Equations

, Volume 4, Issue 1, pp 59–87

The Euler and Weierstrass conditions for nonsmooth variational problems


  • A. D. Ioffe
    • Department of MathematicsTechnion
  • R. T. Rockafellar
    • Departments of Mathematics and Applied MathematicsUniversity of Washington

DOI: 10.1007/BF01322309

Cite this article as:
Ioffe, A.D. & Rockafellar, R.T. Calc. Var (1996) 4: 59. doi:10.1007/BF01322309


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.

Mathematics subject classifications (1991)


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© Springer-Verlag 1996