Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.

Is the mapping \begin{equation} \begin{array}{rcl} C^0([0,T],C^1(\bar{\Omega})) & \to & L^2([0,T],H^1(\Omega))\\ \theta & \mapsto & f \circ (\mathcal{I} + \theta) \end{array} \end{equation}

continuous ?

EDIT : $\mathcal{I}$ is the identity mapping of $\mathbb{R}^n$.