I have this functional defined from a Hilbert space $H$, $J\colon H\rightarrow \mathbb{R}$ defined by:
$$ J(u)=\frac12 \|u\|^2-\int_0^1(A(su),u) ds $$ where $A\colon H\rightarrow H$ is a potential operator.
and I want to prove that $$J'(u)=u-A(u)$$
where $(.,.)$ is the scalair prudact on $H$.
How to do please ?
Edit1: I tried to see $\displaystyle\lim_{t\rightarrow 0} \frac{J(u+tv)-J(u)}{t}=(u,v)-\lim_{t\rightarrow 0}\frac1t\int_0^1 (A(s(u+tv)),u+tv)-(A(su),u) ds=(u,v)-\lim_{t\rightarrow0} \frac1t\int_0^1(A(s(u+tv))-A(su),tv)ds$
But i don't know how to continue?
Thank you.
Formally, it is trivial. Assume that $A=dB$, according to the definition of a potential operator. Then $$ \frac{d}{ds}B(su)=dB(su)(u)=\langle A(su),u \rangle. $$ Therefore $$ J(u)=\frac{1}{2}\|u\|^2 - B(u), $$ assuming that $B(0)=0$. As a consequence, $\nabla J(u)=u-A(u)$. Of course we need to justify these manipulations, but I guess that you can add the necessary assumptions on $A$.