Let $\Omega \subset \mathbb{R}^{2}$ be a bounded and connected domain with a smooth boundary $\Gamma$. Furthermore, let $M$ be a matrix-valued function, where the entries $m_{ij}$ are smooth and bounded for each $i,j \in \{ 1,2\}$. We also assume that $M$ is symmetric, i.e., $m_{12}=m_{21}$. Furthermore, we assume that $M$ is indefinite, i.e. it has one positive and one negative eigenvalue for every $(x,y) \in \Omega$ and at least one zero eigenvalue for $(x,y) \in \Gamma$.
Then I want to show that the PDE
\begin{align}
\nabla \cdot \left( M \nabla u\right)=0, \qquad & \text{ in } \Omega, \\
u=g \qquad & \text{ on } \Gamma,
\end{align}
has a unique solution, where $g$ is some arbitrary smooth function.
This is a hyperbolic PDE in $\Omega$ and parabolic on the boundary $\Gamma$. This is clear since, by using the the product rule and $u_{xy}=u_{yx}$ we get, $$ \nabla \cdot \left( M \nabla u\right) = m_{11}u_{xx} + m_{12}u_{xy} + m_{22}u_{yy} + \text{lower order terms}$$ Then $M$ is hyperbolic if $m_{12}^{2}-m_{11}m_{22}>0$. Which holds since,
$M$ is indefinite $\implies$ $\operatorname{det}(M) < 0$ $\implies$ $m_{11}m_{22}-m_{12}^{2}<0$ $\Leftrightarrow$ $m_{12}^{2}-m_{11}m_{22}>0.$
Solving the special case for $g=0$ (is $u=0$ the only solution), would already be helpful for me. There is extensive literature for similar problems when $M$ is positive (non-negative) PDE is (sub)-elliptic. I assume there should be some literature on my hyperbolic problem as well, however, I could not find any.