 Solution

Choose smooth boundary data ${\phi }_j$ converging uniformly to $\phi$, and let $v_j$ be their harmonic extensions. The maximum principle for harmonic functions givesso $v_j$ converges uniformly on $\overline{B}$ to a continuous function $v$ with boundary value $\phi$. Interior derivative estimates make the convergence smooth on compact subsets of $B$, hence $v$ is harmonic there. Uniqueness follows by applying the maximum principle to the difference of two solutions.