Define $$ Lu:=-\partial_j(a_{ij}\partial_iu)+b_i\partial_iu+cu $$ where $a_{ij}$, $b_i$, and $c$ are all of $C^\infty(\bar \Omega)$ and we assume $\Omega$ is open bounded in $R^N$, smooth boundary, $N\geq 2$. Also, here $A=(a_{ij})$ satisfies uniformly elliptic condition, i.e., $A\xi\xi\geq\theta \,|\xi|^2$
We consider the standard PDE \begin{cases} Lu=f&x\in\Omega\\ u=0&x\in\partial\Omega \end{cases} Then by regularity, we have if $f\in C^\infty(\bar \Omega)$, then $u\in C^\infty(\bar \Omega)$ as well. However, if in addition we assume that $f$ is analytic, could I have $u$ is analytic as well? If yes, where can I find a proof?
Secondly, unless we are in one dimension, we know that if $f\in C^2(\bar\Omega)$, we may not have $u\in C^4(\bar\Omega)$(this is why we need Sobolev and Holder space). But could anybody provide me a counterexample? Thank you!
For your first question, the answer can be found in paper [1] and references therein.
The answer for your second question can be found here.
[1]: Morrey, C. B., Jr.; Nirenberg, L. On the analyticity of the solutions of linear elliptic systems of partial differential equations. Comm. Pure Appl. Math. 10 (1957), 271–290.