There are good books (like Evans) for strongly elliptic second order linear PDE. I want to learn about weakly elliptic PDE (of any order). Are there any good books for the same? I am very curious as to how one could hope to find weak solutions for such PDE. I mean, the only method I know of is the Lax-milgram theorem + Fredholm alternative for compact operators. But the estimates for Lax-miligram don't work for weak ellipticty, do they? (By weak ellipticity I mean the principal symbol is an isomorphism. It isn't necessarily positive definite).
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$\begingroup$ Can you give a definition of "weakly elliptic"? $\endgroup$– Nate EldredgeCommented Apr 22, 2010 at 19:32
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$\begingroup$ Look for books and expository articles on pseudodifferential operators. I'm way of date, but there are, for example, books by Michael Taylor, Francois Treves, and many others. A nice one is by Chazarain and Piriou. $\endgroup$– Deane YangCommented Apr 22, 2010 at 20:47
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Quote:
Weak ellipticity is nevertheless strong enough for the Fredholm Alternative, Schauder estimates, and the Atiyah-Singer Index Theorem. On the other hand, we need strong ellipticity for the Maximum Principle, and to guarantee that the eigenvalues are discrete, and their only limit point is infinity.
from
http://en.wikipedia.org/wiki/Elliptic_operator
Gilbarg, D.; Trudinger, Neil (1983), Elliptic Partial Differential Equations of Second Order, New York: Springer, ISBN 3-540-41160-7
If you find that you are really serious about this consider contacting David R. Adams at the University of Kentucky, Lexington.
answered Apr 22, 2010 at 19:38