Suppose we have a harmonic function $u$ inside the half ball $B_+ \subset \mathbb R^n$ such that $u \in C(\bar B_+)$ and $u$ is $0$ when restricted to the flat part of $\partial B_+$. Then the Schwarz reflection principle states that we can extend $u$ to the whole ball in a harmonic ``way". This is very easy to prove, since we only need to check that the extension is harmonic in $\partial B_+ \cap \partial B_-$ and we can do so by checking that it satisfies the mean value property for harmonic functions.
Suppose $A(x)$ is a $n \times n$ symmetric, real matricial function with $C^\infty$ coefficients (or any regularity you may want) and such that its biggest and smallest eigenvalues are always in the interval $[\lambda, \Lambda] \subset (0, \infty)$ (uniformly elliptic).
My question is whether there exists some domain $\Omega_+$ such that we can do something similar when $u$ solves an elliptic PDE of the form $$ \operatorname{div}(A(x)\nabla u(x))=0$$ in $\Omega_+$? (assuming I can define $A(x)$ in $\Omega_-$ any way I want with the only constraint that $A(x)$ keeps the same regularity)
My intuition tells me that it doesn't hold true with a half ball, but that maybe it does with a shape $\Omega_+$ which should be ``close" to a ball (although maybe very small, with size depending somehow on $\Vert a_{ij}'(x) \Vert_\infty$). But in any case, now I don't have a powerful tool such as the mean value property (or do I?), so I don't know how to proceed.
Does anyone know some reference on this?