Regularity up to the boundary Let $L$ be a second order linear elliptic differential operator on an open bounded subset $U\subset \mathbb R^n$, with smooth uniformly bounded coefficients. Suppose the boundary of $U$ is $C^\infty$. Suppose $f\in C_c^\infty(U)$ ($f$ is smooth and has compact support in $U$). Must there exist a solution $u$ to the PDE $Lu = f$, $u|_{\partial U} = 0$ such that $u$ extends to be $C^\infty$ on the closure $\bar U$?

Yes, this is the Schauder existence theory (see, for example, Gilbarg and Trudinger, Section 6.3 or so, or if that is not readily available, the Wikipedia article http://en.wikipedia.org/wiki/Schauder_estimates has a good summary). 
Applying it once will give you $C^{2,\alpha}$ estimates. Then you subsequently apply the theory to the first derivatives, recognizing that differentiating the equation gives you 2nd order elliptic operator in the first derivative, and so on for higher order terms. 
