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



Reference to the Existence and Uniqueness of the PDE system
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.


