# reference for system of first order partial differential equations

I want to know something from basic to recent results on solutions of first order partial differential equations (on manifold, or simply R^n), like existence, uniqueness and regularity of (weak) solutions (locally and globally), could any one recommend some references? I know basic stuff of analysis and differential equations. Thanks.

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There is only a limited amount of general theory for systems of PDE's, but a lot of results for specific types of systems, especially those arising from specific applications. So it would be best if you could describe in more detail what kind of systems you are looking at (linear or nonlinear; elliptic, hyperbolic, or neither) and the context in which these systems arise. The application often dictates the best approach to studying the systems of PDE's. –  Deane Yang Apr 11 '10 at 2:32
Thanks, you two guys. I didn't have a specific problem. I had a chat with one friend on differential equation a couple of days ago, and he said most PDE theory concerns various specific types of second order PDEs (as mentioned by Deane Yang), and here I'm just wondering how much do we know about linear partial differential equations of order one in general (or for some specific type). –  user1832 Apr 11 '10 at 13:09
Usually, broad `questions' like "Let tell me all about X" are quite hard to answer sensibly... as the FAQ (mathoverflow.net/faq) puts it, «MathOverflow is not the appropriate place to ask somebody to write an expository article for you.» :) –  Mariano Suárez-Alvarez Apr 11 '10 at 14:26
First order systems can also be classified into types including elliptic and hyperbolic. Each type exhibits significantly different behavior and requires different techniques to solve. In general, first order systems are studied using mainly integral estimates in contrast to scalar second order equations, which can also be studied using pointwise bounds obtained by the maximum principle. –  Deane Yang Apr 11 '10 at 16:20
In a different direction, the algebraic structure of an overdetermined system of PDE's can be quite intricate and fascinating to some people. The main theorems in this subject are the Cartan-Kahler and Cartan-Kuranishi theorems, but Spencer, Guillemin, Sternberg, and Goldschmidt did a lot of interesting work redoing everything in a cohomological framework. –  Deane Yang Apr 11 '10 at 16:23