Does elliptic regularity guarantee analytic solutions? 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).
 A: While probably not the fastest approach I think that Hörmander: The analysis of linear partial differential equations, IX:thm 9.5.1 seems to give a (positive) answer to your question. It is overkill in the sense that it gives you a microlocal statement telling you that for $Pu=f$, $u$ is analytic in the same directions as $f$ is.
A: Also: there is a classical result due to Charles Morrey, "Analyticity of the solutions of analytic non-linear elliptic systems of partial differential equations", that says that if $F(x,u,\nabla u,\nabla^2 u,...)$ is analytic in its arguments and elliptic then the solution of $F(x,u,\nabla u, \nabla^2 u,...)=0$ will be as well. (It actually goes one step further to deal with systems, but the notion of ellipticity is complicated to explain.) This result generalizes work done since the early 1900's; references can be found in Fritz John's (and two other author's I can't recall) pde book.  
A: The keyword is "analytic hypoellipticity".
Indeed, the answer to your question is, apparently unfortunately, affirmative. This is a result of Petrowsky [Petrowsky, I. G. Sur l'analyticité des solutions des systèmes d'équations différentielles. (French) Rec. Math. N. S. [Mat. Sbornik]  5(47),  (1939). 3--70. MR0001425  (1,236b)] Cf. also [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. MR0089334  (19,654b)]
That theorem, in the case of constant coefficients, was one of the peaks of my undergraduate education :)
