given the set $$ M:=\{u\in H^2(\Omega):\int_{\Omega}u=m\,\} $$ $m\in \mathbb{R}$, $\Omega $ is a bounded piecewise smooth domain in $\mathbb{R}^n$.
also denote by $u(t)$ a map: $u(t):(0,T)\to M$
and given a functional $E:\mathbb{R}\to M\to \mathbb{R}$ by $E[u(t)]$.
I want use the chain rule to differentiate $\frac{d}{dt}E=\nabla E\cdot u_t$,
but I am not sure that $M$ is Banach space (so I can't use Frechet derivative),
so what is the correct way of using the chain rule, namely , what is the definition of $\nabla E$ on M (parallel to Frechet derivative)?
Does M is some manifold?
thank you