Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading coefficient matrix $A(x)$ is only positive definite, rather than strong elliptic type $$A(x)\xi\cdot\xi\geq\lambda|\xi|^2,\quad\forall\xi\in\mathbb{R}^{mn},\quad a.e. x\in\Omega$$
Any answer and reference will be appreciated!
In 1948 A.V. Bitsadze, “On unique solvability of the Dirichlet problem for elliptic partial differential equations,”Uspekhi Mat. Nauk [Russian Math. Surveys],3, No. 6, 211–212,
constructed an elliptic equation with complex coefficients $$ Lu=\frac{\partial^2 u}{\partial x^2}+2i \frac{\partial^2 u}{\partial x\partial y}+\frac{\partial^2 u}{\partial x^2}=0 $$ for which the Dirichlet problem in the unit circle $D=\{x^2+y^2<1\}$, $$ Lu=0 \text{ in } D,\quad u|_{\partial D}=0, $$ is neither Fredholm, nor Noetherian. Namely, there are infinitely many solutions of this problem of the form $u(z)=f(z)(1-|z|^2)\,$ where $f$ is an analytic function in $\bar D$.
In the real form it is a uniformly elliptic system $$ \frac{\partial^2 u_1}{\partial x^2}-2 \frac{\partial^2 u_2}{\partial x\partial y}-\frac{\partial^2 u_1}{\partial y^2}=0, $$ $$ \frac{\partial^2 u_2}{\partial x^2}+2 \frac{\partial^2 u_1}{\partial x\partial y}-\frac{\partial^2 u_2}{\partial y^2}=0. $$ The notion of a strong elliptic system was introduced exactly to get the case where the corresponding operators are still Noetherian.
To complement Andrew's answer, Fredholm alternative holds for properly elliptic systems (with complementing boundary conditions). In 3 or more dimensions, any elliptic system in the sense of Douglis-Nirenberg is properly elliptic. Note that this is much more general than strongly elliptic systems. Strongly elliptic systems (with appropriate boundary conditions) have index 0, while a general elliptic system can have a nonzero index. Please see Agmon-Douglis-Nirenberg '64.
