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Ricardo Andrade
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Igor Khavkine
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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 $B=\{\{0,0,1,0\},\{0,0,0,1\},\{-1,0,0,0\},\{0,-1,0,0\}\}$ so $$B=\begin{pmatrix} 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ -1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 \end{pmatrix}$$ 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.

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 $B=\{\{0,0,1,0\},\{0,0,0,1\},\{-1,0,0,0\},\{0,-1,0,0\}\}$ 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.

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.

Putting math symbols in $$.
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edit approved Nov 9, 2013 at 22:49
user42090
edit approved Nov 9, 2013 at 22:49
user42090

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

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

Here B={{0,0,1,0},{0,0,0,1},{-1,0,0,0},{0,-1,0,0}}$B=\{\{0,0,1,0\},\{0,0,0,1\},\{-1,0,0,0\},\{0,-1,0,0\}\}$ so has doubly degenerate eigenvalues \pm i$\pm i$. 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.

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={{0,0,1,0},{0,0,0,1},{-1,0,0,0},{0,-1,0,0}} 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.

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 $B=\{\{0,0,1,0\},\{0,0,0,1\},\{-1,0,0,0\},\{0,-1,0,0\}\}$ 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.

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user42587
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