tes the proof.

The term compatibility condition used above for the assumption for $g$ comes from the observation that since the solution $\psi$ is to satisfy and we have $\psi\in C(I,\tilde{W}^2(G\times S))$, it follows that $\psi(E_m)=0$, and therefore $0=\psi(E_m)|_{\Gamma_-}=g(E_m)$.

Strictly speaking, in the proof of Theorem \[evoth1\] above we did not fully address the claim that $u\in C(I,\tilde{W}^2_{-,0}(G\times S))$, since, as pointed out in Remark \[re:evoth\_norm\], the common domain $D=D(A_C(E))=\tilde{W}^2_{-,0}(G\times S)$ of the operators $A_C(E)$, $E\in I$, is to be equipped with the graph norm of $A_C(0)$ when applying Theorem \[evoth\]. However, the norm ${\left\Vert \cdot\right\Vert}_{\tilde{W}^2_{-,0}(G\times S)}$ of $\tilde{W}^2_{-,0}(G\times S)$ is equivalent to the graph norm ${\left\Vert \cdot\right\Vert}_{A_C(E)}$ in $D=\tilde{W}^2_{-,0}(G\times S)$, for every $E\in I$, and in particular for $E=0$. This can be readily seen from the estimates, for $\phi\in D$, $${\left\Vert A_C(E) \phi\right\Vert}_{L^2(G\times S)}\leq{} & \frac{1}{\kappa} \big({\left\Vert \omega\cdot\nabla_x\phi\right\Vert}_{L^2(G\times S)} + C_2{\left\Vert \phi\right\Vert}_{L^2(G\times S)}\big), \\
{\left\Vert \omega\cdot\nabla_x\phi\right\Vert}_{L^2(G\times S)} \leq{} & C_1{\left\Vert A_C(E)\phi\right\Vert}_{L^2(G\times S)} + C_2{\left\Vert \phi\right\Vert}_{L^2(G\times S)},$$ where $$C_1:={\left\Vert \tilde{S}(E)\right\Vert}_{L^\infty},\quad
C_2:=C C_1 + {\left\Vert \tilde{\Sigma}(E)\right\Vert}_{L^\infty} + {\left\Vert {\frac{\partial \tilde{S}}{\partial E}}(E)\right\Vert}_{L^\infty} + {\left\Vert \tilde{K}(E)\right\Vert},$$ and we have used a short hand notation $L^\infty$ for $L^\infty(G\times S)$. Notice also that ${\left\Vert \gamma_-(\phi)\right\Vert}_{T^2(\Gamma_-)}=0$ when $\phi\in \tilde{W}^2_{-,0}(G\times S)$.

Let $H^m(I,X),\ m\in{\mathbb{N}}_0$ be the Sobolev space for Hilbert space valued functions $I\to X$. We have the following corollary.

\[evothco\] Suppose that the assumptions , , , and , and of Theo