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.

ellipticboundary value conditions, and $\Omega$ is compact. $\endgroup$ – Liviu Nicolaescu Jun 6 '13 at 10:00