= Solution
Choose smooth boundary data $\varphi_j$ converging uniformly to $\varphi$, and let $v_j$ be their harmonic extensions. The <maximum principle for harmonic functions> gives
$$
\|v_j-v_k\|_{C^0(\overline B)}
\leq\|\varphi_j-\varphi_k\|_{C^0(\partial B)},
$$
so $v_j$ converges uniformly on $\overline B$ to a continuous function $v$ with boundary value $\varphi$. 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.
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