It is known that if $U$ is a bounded smooth domain in $\mathbb{R}^n$ than for every $f\in C^{\infty}(\bar{U})$ the solutions to the problem $$ Lu=f $$ where $L$ is uniformly elliptic operator with smooth coefficients in $\bar{U}$, of divergence form. are smooth.
What happen if $U$ is not smooth, only piecwise smooth , e.g., a square in $\mathbb{R}^2$?
Can someone give a reference?
thanks