I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution or a hint. Hopefully, you can help me. Thanks a lot already.
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., $m_{12}=m_{21}$. Now to the interesting part. We assume that $M$ is not positive definite, it is negative definite in some region and indefinite in another region.
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.
Now let me explain my thinking so far. If $M$ were entirely positive (or negative) definite this problem would be easy, as the existence and uniqueness then follows from Lax-Milgram. However, since $M$ is sometimes indefinite it becomes more difficult. I tried to solve the problem first for $g=0$. Clearly, in this case, $u=0$ is a trivial solution. However is it the only one? I tried to separate $\Omega$ into parts where $M$ is negative (there the problem is elliptic) and where $M$ is not definite, i.e., has one negative and one positive eigenvalue (here the problem is hyperbolic). Maybe there is a non-trivial solution in the hyperbolic area that we can "glue" together with $u=0$ on the boundary and in the area where $M$ is negative definite. However, I could not show the existence of another non-trivial solution in the area where $M$ is hyperbolic.
Now my question. Does this PDE have a unique weak solution? And is there a non-zero solution for the case $g=0$? And would I show either of the two things?
Note on the definiteness of the problem. The problem is hyperbolic where $M$ is indefinite, and elliptic where it is (negative) definite.
Using the the product rule and $u_{xy}=u_{yx}$ yields, $$ \nabla \cdot \left( M \nabla u\right) = m_{11}u_{xx} + m_{12}u_{xy} + m_{22}u_{yy} + \text{lower order terms}$$ Then $M$ is hyperbolic if $m_{12}^{2}-m_{11}m_{22}>0$.
This holds since, $M$ is indefinite $\implies$ $\operatorname{det}(M) < 0$ $\implies$ $m_{11}m_{22}-m_{12}^{2}<0$ $\Leftrightarrow$ $m_{12}^{2}-m_{11}m_{22}>0$
