I am working with Schwarz’ Hodge Decomposition - A Method for Solving Boundary Problems.
It is stated that the existence and $H^2$ regularity of the Dirichlet potential imply that the kernel of the boundary value problem \begin{equation}\label{dirichlet-potential-boundary-value-problem} \left\{ \begin{aligned} \Delta\,\phi_D &= \eta & \text{ on } M\\[0.5ex] t\,\phi_D &= 0 & \text{ on } \partial M\\[0.5ex] t\,\delta\,\phi_D &= 0 & \text{ on } \partial M. \end{aligned} \right. \end{equation} is $$\ker(\Delta|H^2\Omega^k_{hom})=\mathcal{H}^k_D(M)$$ and its cokernel is $\mathcal{H}^k_D(M)$ too, where $$H^2\Omega^k_{hom}(M)=\{\eta\in H^2\Omega^k(M)\mid t\,\eta=0, t\,\delta\,\eta=0\}$$ and $$\mathcal{H}^k_D(M)=H^1\Omega^k_D(M)\cap \mathcal{H}^k(M)=\{\omega\in H^1\Omega^k(M)\mid t\,\omega=0, d\,\omega=0, \delta\,\omega=0\}$$ the space of harmonic Dirichlet fields.
I don’t quite understand both of this statements.
I can see that $\mathcal{H}^k_D(M)\subset\ \ker(\Delta|H^2\Omega^k_{hom})$. But why does the other direction hold too?
We know that if $\phi_D$ is a Dirichlet potential of $\eta\in L^2\Omega^k(M)$ we have that $\phi_D\in H^2\Omega^k(M)$ and that $0=\eta\in \mathcal{H}^k_D(M)^\perp\subset L^2\Omega^k(M) $ implies that $\phi_D\in H^1\Omega^k_D(M)\cap\mathcal{H}^k_D(M)^\perp$ such that $$<<d\phi_D, d\xi>>+ <<\delta\phi_D, \delta\xi>>= 0 \qquad \forall \xi\in H^1\Omega^k_D(M).$$
I fail to see how $d\delta\phi_D+\delta d\phi_D=0$, $t\phi_D=0, t\delta\phi_D=0, t d\phi_D=0$ imply that $d\phi_D=0$ and $\delta\phi_D=0$.
Also why is the cokernel also the space of Dirichlet fields?