Let $D$ be an elliptic operator on $\mathbb{R}^n$ with real analytic coefficients. Must its solutions also be real analytic? If not, are there any helpful supplementary assumptions? Standard Sobolev methods seem useless here, and I can't find any mention of this question in my PDE books.

I began thinking about this because I overheard someone using elliptic regularity to explain why holomorphic functions are smooth. Aside from the fact that I find that explanation to be in poor mathematical taste (I regard the beautiful regularity properties of holmorphic functions as fundamentally topological phenomena), it occurred to me that standard elliptic theory falls short of exhibiting a holomorphic function as the limit of its Taylor series. So I'm left wondering if this is an actual limitation of elliptic regularity which could vindicate and entrench my topological bias.

In the unfortunate event of an affirmative answer to my question, I would be greatly interested in geometric applications (if any).