TITLE: Fredholm alternative result for general elliptic system?
QUESTION [7 upvotes]: 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!

REPLY [2 votes]: 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.