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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.

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