View source for Du Bois-Reymond lemma

If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340501.png" /> is a continuous function on the segment <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340502.png" /> and if for all continuously-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340503.png" /> which vanish at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340504.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340505.png" /> the relation

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340506.png" /></td> </tr></table>

is valid, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340507.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d034/d034050/d0340508.png" />. Formulated by P. du Bois-Reymond [[#References|[1]]].

The du Bois-Reymond lemma is employed in the calculus of variations to derive the [[Euler equation|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.

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



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. [S.V. Fomin] Fomine,   "Commande optimale" , MIR  (1982)  (Translated from Russian)</TD></TR></table>