Suppose $S(r)$ is set parametrised by $r \in [0,T]$. Let $\phi_t^s : H^1(S(t)) \to H^1(S(s))$ is a linear homeomorphism.
Suppose $\lVert \cdot \rVert_{H^1(S(t))}$ and $\lVert \phi_t^s(\cdot) \rVert_{H^1(S(s))}$ are equivalent norms on $H^1(S(t))$. So their dual norms are also equivalent.
The norm of $f \in H^{-1}(S(s))$ can be given as $$\sup_{w \in H^1(S(s))} \frac{\langle f, w \rangle_{H^{-1}(S(s)), H^1(S(s))}}{\lVert w\rVert_{H^1(S(s))}} = \sup_{v \in H^1(S(t))} \frac{\langle (\phi_t^s)^*f, v \rangle_{H^{-1}(S(t)), H^1(S(t))}}{\lVert \phi_t^sv\rVert_{H^1(S(s))}}$$ where the star denotes the adjoint.
I can derive this by just using the fact that \phi_t^s is a homeomorphism and the adjoint identity. Do I need any equivalence of dual norms at all?
Suppose that $$\lVert \phi_t^sv\rVert_{H^1(S(s))} \leq C\lVert v \rVert_{H^1(S(t))}$$
This, and the equivalence of $H^1$ norms implies $$\lVert (\phi_t^s)^*\rVert_{\mathcal{L}(H^{-1}(S(s)), H^{-1}(S(t))} \leq C$$
I still don't need any equivalence of norms to get this. Can somebody show me how to do it correctly?
We don't need the assumption that the norm are equivalent or that $\lVert\phi_t^s v\rVert_{H^1(S(s))}\leq C\lVert v\rVert_{H^1(S(t))}$, as it's already implied by the property of linearity and homeomorphism. Indeed, if $L\colon E\to F$ is linear and continuous, where $E$ and $F$ are normed spaces, then we can find $C>0$ such that $$\lVert Lx\rVert_F\leq C\lVert x\rVert,\quad x\in E.$$
To see the equality between the two suprema, we just need the fact that $\phi_t^s$ is a bijective correspondance.