Du Bois-Reymond lemma

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If $N$ is a continuous function on the segment $[x_1,x_2]$ and if for all continuously-differentiable functions $\eta$ which vanish at $x=x_1$, $x=x_2$ the relation


is valid, then $N=\text{const}$ on $[x_1,x_2]$. Formulated by P. du Bois-Reymond [1].

The du Bois-Reymond lemma is employed in the calculus of variations to derive the Euler equation in its integral form. In this proof it is not necessary to assume that the extremum of the functional is attained on a twice-differentiable curve; the assumption of continuous differentiability is sufficient.


[1] P. du Bois-Reymond, "Erläuterungen zu der Anfangsgründen der Variationsrechnung" Math. Ann. , 15 (1879) pp. 283–314
[2] M.A. Lavrent'ev, L.A. Lyusternik, "A course in variational calculus" , Moscow-Leningrad (1950) (In Russian)



[a1] S. [S.V. Fomin] Fomine, "Commande optimale" , MIR (1982) (Translated from Russian)
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Du Bois-Reymond lemma. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by I.B. Vapnyarskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article