Let
- $T>0$
- $d\in\mathbb N$
- $\Omega\subseteq\mathbb R^d$ be open and $\Lambda\subseteq\mathbb R^d$ be bounded and open with $\overline\Lambda\subseteq\Omega$
- $u\in C^{1,\:2}([0,T]\times\Omega,\mathbb R^d)$ and $$\left.\tilde u(t):=u(t,\;\cdot\;)\right|_{\overline\Lambda}\;\;\;\text{for }t\in[0,T]$$
Is $\tilde u$ Fréchet differentiable?
By definition, Fréchet differentiability depends on the involved Banach spaces. Since ${\rm D}^\alpha\tilde u(t)$ is continuous in $\overline\Lambda$ and $\overline\Lambda$ is compact, ${\rm D}^\alpha\tilde u(t)$ is bounded in $\Lambda$ and hence $${\rm D}^\alpha\tilde u(t)\in L^p(\Lambda,\mathbb R^d)\;\;\;\text{for all }p\in[1,\infty]\tag 1$$ for all $|\alpha|\le 2$ and $t\in[0,T]$. Since $\tilde u(t)\in C^2(\Omega)$, we can even conclude that $$\tilde u(t)\in W^{2,\:p}(\Lambda,\mathbb R^d)\;\;\;\text{for all }p\in[1,\infty]\;.\tag 2$$ for all $t\in[0,T]$. So, we might consider (for example) $$(0,T)\to L^p(\Lambda,\mathbb R^d)\;,\;\;\;t\mapsto\tilde u(t)\tag 3$$ or $$(0,T)\to W^{2,\:p}(\Lambda,\mathbb R^d)\;,\;\;\;t\mapsto\tilde u(t)\;.\tag 4$$
The question is: Are $(3)$ and/or $(4)$ Fréchet differentiable?