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Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear or nonlinear elliptic system. Further more, may I say that we can parallelly extened the $L^2$-theory of the second-order scalar elliptic equations to the corresponding $\mathbb{L}^2$-theory of the second-order elliptic systems? Any answer and reference will be appreciated!

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The original papers of Agmon, Douglis and Nirenberg are a good place to start. –  Michael Renardy Jun 4 '13 at 13:52
    
@Michael Renardy: Thank you for your answer! Can you provide some crash guide, lecture notes or good books on elliptic systems for me. Because I have to learn the theory ,and eventually applied them to the system I concerned. –  A.Hoo Jun 4 '13 at 14:09

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All you mentioned can be extended to elliptic systems. The only things you should be careful about are the maximum principles and DeGiorgi-Nash-Moser type regularity results. This means for linear systems the theory is very satisfactory. You can start with strongly elliptic systems and the variational approach to them, because the theory is much simpler and it covers a lot of important systems already. A good place to start would be McLean's book, Wloka's book, and perhaps Nirenberg's article. Wloka's book also contains a comprehensive introduction to the general elliptic theory, in the style of Agmon-Douglis-Nirenberg.

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@timur: Thank you! :) –  A.Hoo Jun 4 '13 at 15:05
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@timur: Now I have known that the Fredholm alternative result is valid for strong elliptic system. However, I wonder is it still valid for general second-order linear elliptic system (not strong type)? :) –  A.Hoo Jun 6 '13 at 6:40
    
@A.Hoo: Yes, it is valid, for a subclass called properly elliptic systems. This class is huge, in particular much bigger than the class of strongly elliptic systems. There is a requirement to be placed on the boundary condition, called the Lopatinsky-Shapiro condition. For example, the Dirichlet condition will do. You should keep in mind that strongly elliptic systems are Fredholm of index 0, while general elliptic systems can have nonzero index. Computing this index on closed manifolds in terms of topological information is the goal behind the Atiyah-Singer index theorem. –  timur Jun 6 '13 at 15:55

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