I spent a long morning at the library trying to understand why it so happens that the Cauchy problem for elliptic partial differential equations is not well posed. I still don't - I understand that it is, but what is the geometrical property of elliptic equations that makes it not well-posed, in contrast to parabolic or hyperbolic equations?
In the process I was not impressed by the creativity shown in textbooks. There are some good books out there - but the majority of them are isomorphic to each other. Most are well-written, a few stand out. But mostly they present the same topics in a different order, with more or less space devoted to each of them. Every preface should answer: "Why is this book different from all the other books?" Unfortunately, it seems that the question is not on most publishers' minds.
The good side is that there are plenty of open problems, even in areas that have already been studied a million times. Ask good questions, don't try to read all the literature, and have fun!
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As far as I know, it is not so easy to prove that the Cauchy problem is ill-posed for an elliptic PDE. You should probably just try to figure out why this is true for the standard Laplacian on Euclidean space. And unfortunately I don't know of a specific reference for this, either.
Note that this is a fact in analysis and does not follow by geometric or algebra. The Cauchy problem for an elliptic PDE *is* well-posed both formally and in the real analytic category (because you can solve it using power series). I believe there is a proof of ill-posedness in the book by Chazarain and Piriou, "Introduction to the theory of linear partial differential equations".
Thanks! Good point. The standard explanation is an example by Hadamard. Easy to find in PDE textbooks, it just shows lack of continuity for a Laplace equation. There is no reason why more complicated elliptic equations would exhibit continuity.
You have a good point: it's a fact of analysis. It may or may not have a geometric explanation. It would help me understand if one existed, but otherwise it's just a fact of life. Also, as you mention "well-posedness" has different meanings in different contexts.
Thanks again!
Lack of (real) (bi)characteristics is the geometrical "explaination"
for the ill-posedness of Cauchy problems for elliptic PDEs.
Having said this, I have to add that this is a rather empty statement,
since lack of characteristics cannot be used for a proof of
ill-posedness. In this sense, your question is ill-posed. It is like
asking why a non-elephant has no trunk.
One has to remember, though, that there are circuntances where Cauchy
problems for elliptic PDEs do have at least local solutions. One find
various theorems about this in Avner Friedman "Partial Differential
Equations of Parabolic Type", Dover (1992).
If I correctly understood the statements made there, they are not only
for the "real analytic category" (Oh, how I hate this category
lingo...), i.e. they go beyond the old Cauchy-Kovalevskaya Theorem.
There is also Yoshio Kato, "The Poisson Kernels and the Cauchy Problem
for Elliptic Equations with Analytic Coefficients" Transactions of the
American Mathematicalal Society, Vol. 144, (1969). See:
www.jstor.org/stable/1995284
But it is only for the "real analytic category".
Is "bicharacteristics" a standard term? I found the references a little confusing. What exactly does it mean?
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