A Note on the Du Bois-reymond Equations in the Calculus of Variations

satisfies the du Bois-Reymond form of the Euler equations. The principal purpose of the present note is to give an alternate proof of these results of McShane and Tonelli. In addition to being simpler in detail than the previous proofs, the differentiability theorems of §§2 and 3 involve weaker hypotheses than the corresponding theorems of McShane and Tonelli, both in regard to conditions on the integrand function and in regard to the class of arcs considered. Basically, the present proof is intimately related to the proof of the fundamental lemma as given by Bliss [l, pp. 20-21].