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Consider the energy function $$F(u)=\int_\Omega \frac{1}{2}|\nabla u|^2+fu dx,$$ where $u\in W^{1,2}(\Omega)$, $\Omega$ is a bounded domain in $\mathbb{R}^n$, and $f\in L_2(\Omega)$. I'd like to find the first order functional derivative of $F$, and determine the Fréchet and Gâteaux differentiability of $F$.

Here I present my work so far:

The differential $d F$ due to an infinitestimal change $\delta u$ is given by $$\begin{aligned}d F&=F(u+\delta u)-F(u)=\int_\Omega \frac{1}{2}(|\nabla u+\nabla(\delta u)|^2-|\nabla u|^2)+f\delta ud x\\ &=\int_\Omega\nabla u\cdot\nabla\delta u+f\delta u+\frac{1}{2}|\nabla\delta u|^2d x\\ &=\int_\Omega \nabla\cdot((\nabla u)\delta u)-(\nabla\cdot(\nabla u))\delta u+f\delta u+\frac{1}{2}|\nabla\delta u|^2dx. \end{aligned}$$ Therefore the first order functional derivative is $$\frac{\delta F}{\delta u}=-\nabla\cdot(\nabla u)+f=-\Delta u+f.$$ From the above computation we see that $$\begin{aligned}\frac{F(u+\epsilon y)-F(u)}{\epsilon}&=\frac{1}{\epsilon}\int_\Omega \frac{1}{2}(\epsilon(\nabla u\cdot\nabla y))+\epsilon fy+\frac{1}{2}\epsilon^2|\nabla y|^2dx\\ &=\int_\Omega \frac{1}{2}(\nabla u\cdot\nabla y)+fy+\frac{1}{2}\epsilon|\nabla y|^d x\\ &\to \int_\Omega \frac{1}{2}(\nabla u\cdot\nabla y)+fydx. \end{aligned}$$ Hence $F$ is Gâteaux differentiable at $u$, with $$F'(u)y=\int_\Omega \frac{1}{2}(\nabla u\cdot\nabla y)+fydx.$$

Finally for Fréchet differentiability, this is the part I am not sure how to proceed. My idea to this is the following: we have $$F(u+y)-F(u)-L(u)y=\frac{1}{2}\int_\Omega \nabla u\cdot\nabla y+|\nabla y|^2d x=\frac{1}{2}\int_\Omega (\nabla u+\nabla y)\cdot\nabla yd x,$$ where $L$ is the bounded linear functional $$L(u)y=\int_\Omega fudx.$$ $F$ is Fréchet differentiable at $u$ if and only if $$R(u,y):=\frac{\frac{1}{2}\int_\Omega (\nabla u+\nabla y)\cdot\nabla yd x}{\|y\|_{W^{1,2}(\Omega)}}\to 0,\text{ as }y\to 0.$$ When $\nabla u=0$, i.e. when $u=c$ for some constant $c$, we have $$R(u,y)=\frac{\int_\Omega\|\nabla y|^2dx}{2\|y\|_{W^{1,2}(\Omega)}}\leq\frac{\|y\|_{W^{1,2}(\Omega)}^2}{2\|y\|_{W^{1,2}(\Omega)}}\to 0,\text{ as }y\to 0,$$ and in this case the Fréchet derivative is given by $$L(u)y=c\int_\Omega fd x.$$ Then we are left with the case $|\nabla u|\neq 0$. I really didn't have much idea on dealing with this. Any one have some suggestions? Also it will be greatly appreciated if someone can verify my work so far.

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I think the book Partial Differential Equations by Evans can help you. Look at chapter 8.1.2 and and Example 1-3 at Page 456.

In short, the Gâteaux derivative of $F$ are all the weak form of $<\nabla u,\nabla\cdot>+<f,\cdot>$, basically what you got from the first variation, provided that $f$ satisfies some mild assumptions.

Now for Fréchet derivative. For this special example, it is equivalent to Gateaux. To see how, look at theorem $C.1$ in Apendix C for this book by Struwe

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