= Solution
Use the <Poisson integral> to choose a smooth <harmonic function> $G$ on the <unit disk> whose boundary value is $g$. Seek $v$ in the form $v=w+G$, which is the standard <homogenization of Dirichlet boundary data>. Then $w\in H_0^1(\Omega)$ and, since $\Delta G=0$,
$$
-\Delta w+w+A\mathbin\cdot\nabla w=-G-A\mathbin\cdot\nabla G\in L^2(\Omega).
$$
Part (d) gives a unique such $w$, and hence
$$
\boxed{v=w+G}
$$
is the unique <weak solution> with boundary value $g$.
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