Assume $U$ has smooth boundary (or satisfies the exterior cone condition), $a$ and $b$ are Hölder continuous on $\bar{U}$, and $a$ is uniformly elliptic. Then for every Hölder continuous $f : \bar{U} \to \mathbb{R}$ and every continuous $g : \partial U \to \mathbb{R}$, the Dirichlet–Poisson problem has a solution.