I was wondering if there were any recent references dealing with the theory of systems of elliptic PDEs: in particular, someone was telling me about something which sounded like 'Schur complementarity' and reducing a system of elliptic PDE down to one PDE, does anyone know of a recent reference which explains this theory? 


I did try the book by Ladyzhenskaya but there was not that much on systems of PDE, perhaps this is quite an old reference now though.



References for systems of elliptic PDEs
In matrix analysis, the **Schur complement** is an object that you obtain after eliminating a part of the unknowns. It works that way: you have to solve $Mx=b$ where $M$ is a square, invertible matrix. You write the system in block form
$$\begin{pmatrix} A & B \\ C & D \end{pmatrix}\binom{y}{z}=\binom{c}{d},$$
where you are lucky enough that $A$ is invertible too. Then elimination of $y$ yields a system $(D-CA^{-1}B)z=b'$. The Schur complement of $A$ is precisely $D-CA^{-1}B$. Remark that we have the formula $\det M=\det A\cdot\det (D-CA^{-1}B)$.


The situation is similar if you have an elliptic system of PDEs. It can be written in an abstract way as a matrix $M$ of differential operators (including the information of their domains). Then write $M$ blockwise as above, with $A$ an elliptic operator, invertible over its domain. Then the Schur complement is $D-CA^{-1}B$, with domain that of $D$.


