I have a distributional solution to $-\triangle \phi + u \cdot \nabla \phi= f$ in $U \subseteq \mathbb{R}^n$ and $\phi=0$ on $\partial U$.
I have that $U$ is open, bounded, connected, $f:\overline{U}\to \mathbb{R}^+$ is continuous, and $u:U \to \mathbb{R}^n$ is Lipschitz.
I would like to conclude that $\phi \in C^2(U)$ by elliptic regularity. Does anyone have a reference which covers the case I am working in? Extra assumptions on $U$ (like $C^1$ boundary, etc.) or on $u$ would be okay, but I cannot make any extra assumptions on $f$.
Edit: I would like to not assume anything extra on $u$ beyond Lipschitz or possibly $C^1$.