= Fréchet differentiability of a first-order variational integral
{c}
{title2=$DI(u)w=\int(f_uw+f_vw')$}
For $I(u)=\int_a^bf(x,u,u')dx$ on $W^{1,\infty}$, <continuity> of $f,f_u,f_v$ on $[a,b]\times\mathbb R^2$ gives a bounded <Fréchet derivative> in the displayed form. Fixing $u$ and restricting $\|w\|_{W^{1,\infty}}$ to a bounded neighborhood confines all intermediate pairs $(u+tw,u'+tw')$ to a compact range up to null sets. <Uniform continuity> of the <partial derivatives> bounds the integrated first-order remainder by $(b-a)\omega(\|w\|)(\|w\|_\infty+\|w'\|_\infty)$, where $\omega(s)\to0$. This is little-oh of the Sobolev <norm>. The result needs no continuous derivative representative for $u$; essential boundedness suffices.
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