Define the operator $E: H_{per}^{1}\left([0,L]\right) \longrightarrow \mathbb{R}$, given by
$$E(u)=\frac{1}{2}\int_{0}^{L}(u_t^2+u_x^2+\frac{1}{2}(1-u^2)^2)dx,\; \forall \; u \in H_{per}^{1}\left([0,L]\right),$$
where $L>0$ is a fixed constant.
I want to calculate the Fréchet derivative of $E$, for this I started I started by calculating the Gateaux derivative of $E$: we know that $ E $ is differentiable Gateuax if there is $ f \in \left(H_{per}^{1}\left([0,L]\right)\right)' $ (dual space) such that, for $u \in H_{per}^{1}\left([0,L]\right),$
$$v_E:=\lim_{\xi \rightarrow 0} \frac{1}{\xi}\left[E(u+\xi h)-E(u)-f(\xi h)\right]=0,\; \forall \; h \in H_{per}^{1}\left([0,L]\right).$$
I did the math I got to
$$v_E= \int_{0}^{L} (u_t h_t+hu_t+u^3h )\; dx -f(h),$$
but I can’t continue from that point, mainly because I don’t know how to calculate the integral
$$\int_{0}^{L} u_t h_t \; dx.$$
My idea is to calculate the Gateaux derivative (finding such operator $f$) and use the $ E $ continuity to conclude that this derivative coincides with Fréchet's and consequently conclude what I want. How do I proceed?
More details of the space $H_{per}^{1}\left([0,L]\right)$ can be find in this book.
Best Answer
I've no idea what $H^1_{\rm per}[0,L]$ is but it seems to me that you are making heavy weather of a simple and standard calculation. Let $$ E[u]= \frac 1 2\int_0^L (u_t^2+u_x^2 +\frac 12 (1-u^2)^2 dx. $$ then, with $u(x) \to u(x)+\eta(x)$ we have $$ \delta E = \int_0^L \left(u_t \eta_t +u_x \eta_x + (1-u^2)u \eta \right)dx\\ = \int_0^L \left(-u_{tt} \eta -u_{xx} \eta + (1-u^2)u \right)\eta(x) dx\\ = \int_0^L (-u_{tt} -u_{xx} + (1-u^2)) \eta(x) dx $$ We have assumed that $\eta(0)=\eta(L)=0$ so as to allow the integration by parts, and so we can read off the Functional (aka Frechet) derivative to be
$$ \frac{\delta E}{\delta u(x)}= \left(-u_{tt} -u_{xx} + (1-u^2)u \right). $$ I think that the restictions on $u$ and $\eta$ do not need to be very strong $u \in C^2[0,L]$ $\eta\in C^1[0,L]$ should suffice for the manipulation to be legitimate. The extension to the various Banach spaces of Gateaux and Frechet just requires making sure that $\eta$ is in the appropriate dual space.