^\*\_0(\^\*,v)=[**F**]{}\_0\^\*(v),v.

Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), and (\[sda2b\]) are valid. Using integration by parts and Green’s formula we have the following.

For all $\psi,\psi^*\in \tilde {{{\mathcal{}}}H}$ one has \[adjoint7\] \_0\^\*(\^\*,)=\_0(,\^\*), where $\tilde{\bf B}_0(\cdot,\cdot)$ is the bilinear form (\[cosyst5a\]).

The relation is the justification for the term *adjoint problem*. For the existence of solutions of the adjoint problem, we formulate the following result.

\[adjointco1\] Suppose that the assumptions (\[scateh\]), (\[colleh\]), (\[sda2a\]), , (\[sda2b\]) are valid, and that (\[csda3aa\]), (\[csda4aa\]) hold for $c>0$ and for $C$ given in (\[cosyst6a\]). Let $f^*\in L^2(G\times S\times I)^3$ and that $g^*\in T^2(\Gamma_+)^3$. Then the following assertions hold.

\(i) The variational equation \[vareqcoad\] \_0\^\*(\^\*,v)=[**F**]{}\_0\^\*(v)v, has a solution $\tilde{\psi}^*=(\tilde{\psi}^*_1, \tilde{\psi}^*_2, \tilde{\psi}^*_3)\in {{{\mathcal{}}}H}$. Writing $\tilde{\psi}_1^*:=(\psi_1^*,q_1^*)$, $\tilde{\psi}_j^*:=(\psi_j^*,q_j^*,p_{0j}^*,p_{{\rm m}j}^*)$, $j=2,3$, and $\psi^*=(\psi_1^*, \psi_2^*, \psi_3^*)\in L^2(G\times S\times I)^3$, then $\psi^*\in{{{\mathcal{}}}H}_{{\bf P}^*}(G\times S\times I^\circ)$ (see ) is a weak (distributional) solution of the system of equations , , and $\psi^*_1\in W^2(G\times S\times I)$.

\(ii) Suppose that additionally the assumption ${\bf TC}$ holds (p. ). Then a solution $\psi^*$ of the equations , obtained in part (i) is a solution of the problem -.

\(iii) Under the assumptions imposed in part (ii), any solution $\psi^*$ of the problem - that further satisfies \[asscl-aa-co-ad\] \^\*\_[|\_-]{}T\^2(\_-)\^3 (,,E\_[m]{})L\^2(GS)\^3. is unique and obeys the estimate \[adjoint11b\] [\^\*]{}\_[[H]{}]{}( \_[L\^2(GSI)\^3]{}+\_[T\^2(\_+)\^3]{}). (Recall that $C$ is defined in , $c'$ in and that $E_m$ is the cutoff energy.)

Note that if $\psi^*$ is a solution of the problem -, then it is a solution of the varia