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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).

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The example of holomorphic functions (in fact, the statement that an hoomorphic distribution is in fact a smooth function) is given in RUbin's book as an example. –  Mariano Suárez-Alvarez Jul 28 '10 at 5:43
    
Rubin' s book on functional analysis, that is. –  Mariano Suárez-Alvarez Jul 28 '10 at 5:57
    
Rudin's book, that is. :) Example 8.14, to be precise. –  Hans Lundmark Jul 28 '10 at 12:52
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I'm curious about how you propose to show regularity properties of holomorphic functions without appealing to some form of elliptic regularity... –  Rbega Jun 22 '11 at 0:55
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As for a geometric application...While analyticity itself is not so important, one of its consequences is. Namely, the fact that two distinct solutions to some (non-linear) elliptic equation (of an appropriate form) can only agree at a point to finite order. This unique continuation property--which is strictly weaker than analyticity--actually holds for quite a general class of elliptic equations. This comes up, for instance, in the regularity theory for minimal surfaces--specifically in analyzing branch points of minimal surfaces. –  Rbega Jun 22 '11 at 1:16
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3 Answers 3

up vote 5 down vote accepted

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.

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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 :)

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I think it is hypoellipticity and not hipoellipticity so I edited it. –  Torsten Ekedahl Jul 28 '10 at 6:57
    
@Torsten: thanks! –  Mariano Suárez-Alvarez Jul 28 '10 at 17:12
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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.

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