I want to work out the strong form of the following weak description for $f:[0,L]\to\mathbb{R}^2$:
$$ \int_0^L \langle \eta', f \rangle = 0 \quad \text{for all test functions } \eta:[0,L]\to\mathbb{R}^2 \text{ with } \eta(0)=0=\eta(L) \text{ and } \langle \eta', T \rangle=0. $$
Here, $T$ is a fixed smooth function $T:[0,L]\to S^1$. What trips me up is the constraint on $\eta'$ in conjunction with the Dirichlet boundary conditions. Because of this, I do not know how to explicitly construct test functions that I can use to extract the strong form.
I know that, without the constraint $\langle \eta', T \rangle=0$, the answer would simply be that $f\equiv a$ for some constant $a\in\mathbb{R}^2$. I also know that, for the constraint $\langle \eta, T \rangle = 0$, the strong form would be $\langle f', T^\perp \rangle = 0$.
But I cannot find a similar description of $f$ for the constraint $\langle\eta',T\rangle = 0$. Can anyone share pointers on how to tackle this?
Define the two linear spaces $U=\{u:[0,L]\to\mathbb{R}^2: \langle u, T \rangle \equiv 0\}$ and $V=\{v:[0,L]\to\mathbb{R}^2 : \int_0^L v = 0\}$. The derivatives of all test functions span exactly $U\cap V$. The weak form of $f$ defines the orthogonal complement of $U\cap V$ in the sense that $\int_0^L \langle \zeta, f\rangle = 0$ for all $\zeta\in U\cap V$.
The orthogonal complement of an intersection of linear spaces equals the sum of their orthogonal complements. Thus, $f$ satisfies the weak form if and only if $f\in U^\perp + V^\perp$. The space $U^\perp$ is given exactly by functions of the form $gT$, for some $g:[0,L]\to\mathbb{R}^2$. The space $V^\perp$ contains exactly all constant functions, see this answer. Thus, the weak form is satisfied by $f$ if and only if it can be written as $f(s)=g(s)T(s)+b$.