# Reference to the Existence and Uniqueness of the PDE system

I've the following Problem on systems of Partial Differential Equations. I have "$N$" Physical variables. and Finally I form the equation on a bounded domain having regular boundary in $R^d$ ($d=2$ generally)

$\mbox{div}(W_i)=f_i$, $i=1\cdots N$

$W_i =\displaystyle \sum_{i, j=1}^NA_{ij} \cdot\nabla P_j$ with summation indices $j=1\cdots N$ where each $A_{ij}$ is $2\times 2$ non-constant matrix and N unknowns $P_1...P_N$. For $N=1$ based on existing theory of elliptic PDE one can ascertain existence and uniqueness by looking at coefficient matrix.But can someone kindly give any reference to the existence and uniqueness of these kind of problems.And moreover if not then any reference\idea whether existing DN-elliptic systems can be modified to tackle these kind of problems..??

regards ram

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You should $\TeX$ the equations. – timur Aug 23 '12 at 1:23

## 1 Answer

A good keyword here is strongly elliptic systems. There is an original paper by Nirenberg. Also have a look at MacLean's book Strongly elliptic systems and boundary integral operators. Folland's Introduction to PDE has a good treatment too.

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Thanks a lot.And moreover if the A_ij are not constants then is there any way to decouple the system into N PDEs of one variable...??For example by any basis transformation or something like that..!!!!! – Ramu_Dull_Boy Aug 22 '12 at 14:00