 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 $\Delta G=0$,Part (d) gives a unique such $w$, and henceis the unique weak solution with boundary value $g$.