# solving elliptic system of first-order linear PDE's

I am a physicist and while solving linearized Einstein's equations, have come across a system of linear PDE's with $7$ dependent variables and $2$ independent variables. There is a subsystem which decouples from the rest, determining $4$ dependent variables, which written in matrix notation is

$u_{t,i} + B_{i j}u_{x,j}+ C_i=0$.

Here $$B=\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix}$$ so has doubly degenerate eigenvalues $\pm i$. I gather that because these eigenvalues are complex, this is an elliptic system of first-order linear PDE's. Can anybody recommend a good reference which would help me see/explain to me whether and how this elliptic system could be solved analytically?

So far I have come across few references which go into detail about how to solve a system of first-order linear PDE's, in particular ones that are not totally hyperbolic.

• (1) Do you know that you can use LaTeX in the question? Click on 'help' at the top of the page for details. (2) What are the $C_j$? If these terms are linear in $u$ and have constant coefficients, then the best way to get an analytic solution is probably using the Fourier transform. – Igor Khavkine Nov 9 '13 at 21:14
• The C_j have terms linear in u and also source terms that are just functions of independent variables t and x. – user42587 Nov 10 '13 at 2:33

If your equation can be written as $$(\delta_{ij}\partial_t + B_{ij} \partial_x + c_{ij}) u_j = f_i ,$$ where the $c_{ij}$ are constants and $f_i = f_i(t,x)$, then the most general solution can be obtained using the Fourier transform: \begin{gather} (\delta_{ij} (-i\omega) + B_{ij} (ik) + c_{ij}) \tilde{u}_j = A_{ij}(\omega,k) \tilde{u}_j = \tilde{f}_i(\omega,k) , \\ u_j = \int d\omega\,dk\, e^{-i\omega t + ikx} (A^{-1}_{ji}(\omega,k) \tilde{f}_i(\omega,k) + \tilde{g}_j(\omega,k)) , \end{gather} where $\tilde{g}_i(\omega,k)$ is any distribution that is supported only on set where $\det A_{ij}(\omega,k) = 0$, that is, where $A_{ij}$ is singular. Finally, the term $A^{-1}\tilde{f}$ may also have (non-integrable) singularities, so it has to be treated as a distribution as well (think of how the Fourier representation of propagators in quantum field theory is regulated using $+i\epsilon$ or some other prescription).
• There's also a chance that this can be made simpler, since the first order terms appear to uncouple into two 2-by-2 systems, one with $u_1$ and $u_3$ and the other with $u_2$ and $u_4$. If the zero-th order term ($c_{ij}$) also uncouples in the same way, then you can solve each subsystem separately. – Deane Yang Nov 10 '13 at 17:34
• It turns out that I cannot make both matrices $B$ and $c$ constant, i.e. if I want B to be constant, $c \propto 1/t$, and if I let $c$ be constant, $B \sim e^{t}$. It seems the Fourier method is no longer explicity applicable. But is there a good book/reference with some exposition of the Fourier method you mention above, and possibly methods that might apply to my problem? Thanks. – user42587 Nov 11 '13 at 3:34