I've started reading about shape optimization. Most of the concepts I've encountered so far (such as the shape derivatives of domain and boundary integrals and the corresponding) seem to be complex, but turned out to be quite simple. However, I really struggle to wrap my head around the different notions of "derivatives" for a "shape-dependent" function $y$.

The setting is as follows: Let

 - $d\in\mathbb N$;
 - $D\subseteq\mathbb R^d$ be open and $\mathcal A\subseteq 2^D$ with $D\in\mathcal A$;
 - $E_\Omega\subseteq\mathbb R^{\Omega}$ be a $\mathbb R$-Banach space for $\Omega\in\mathcal A$ and $E:=\bigcup_{\Omega\in\mathcal A}E_\Omega$;
 - $y:\mathcal A\to E$ with $$y(\Omega)\in E_\Omega\;\;\;\text{for all }\Omega\in\mathcal A;\tag1$$
 - $\tau>0$ and $T_t:D\to D$ be a $C^1$-diffeomorphism for $t\in[0,\infty);$
 - $v:[0,\tau)\times D\to\mathbb R^d$ be differentiable in the second argument with $$v\left(t,T_t(x)\right)=\frac{\partial T}{\partial t}(t,x)\;\;\;\text{for all }(t,x)\in[0,\tau)\times D;\tag2$$
 - $\Omega\in\mathcal A$ and $\Omega_t:=T_t(\Omega)$ for $t\in[0,\tau)$.

Now the "shape derivative* is defined as follows:

> **Definition 1** (*shape derivative*) Let $Y:[0,\tau)\to E_d$ with $$\left.Y(t)\right|_{\Omega_t}=y(\Omega_t)\;\;\;\text{for all }t\in[0,\tau).$$ Then $y$ is called **shape differentiable at $\Omega$ in direction $v$** if $Y$ is Fréchet differentiable at $0$. In that case, $$y'(\Omega)(v):=\left.Y'(0)\right|_{\Omega}\tag4.$$
(Please note that we most probably need to assume a certain regularity (at least continuity) of the time-dependence of $Y$ (and most probably of $T$ as well). I've omitted them, cause it's part of my question what we need to assume precisely.)

The second definition is given by the "material derivative*:

> **Definition 2** (*material derivative*) $y$ is differentiable at $\Omega$ in the sense of the "material derivative" if if $$y(\Omega_t)\circ\left.T_t\right|_{\Omega}\in E_\Omega\;\;\;\text{for all }t\in[0,\tau)\tag5$$ and $$[0,\tau)\to E_\Omega\;,\;\;\;t\mapsto y(\Omega_t)\circ\left.T_t\right|_{\Omega}\tag6$$ is Fréchet differentiable at $0$. In that case, the material derivative $\dot y(\Omega)(v)$ is defined to be the Fréchet derivative of $(2)$ at $0$.

> **Question 1**: Why is $(3)$ well-defined? In particular, why is it independent of the choice of $Y$? I've only seen a proof of this "fact" for $E_A=L^1(A)$ for all $A\in\mathcal A$ (and most probably under the assumption that the canonical inclusion $L^1(A)\subseteq L^1(B)$ is sufficiently nice for all Borel measurable $A\subseteq B\subseteq D$; don't know if this is always the case).
>
> **Question 2**: What's the relation between both derivatives? Which assumption do we need to impose in order to apply the chain rule to $$[0,\tau)\ni t\mapsto y(\Omega_t)\circ T_t\tag7?$$ I'm sure this would yield a relation.