# The functional derivative , Fréchet and Gâteaux differentiability of the energy operator

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.

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