Let $\Omega\in\mathbb{R}^2$ be a smooth bounded domain and $G(x,y)$ be the Green's function of $-\Delta$ in $\Omega$ with zero Dirichlet condition. Clearly $G(x,y)=-\frac{1}{2\pi}\ln|x-y|-h(x,y)$, where $h$ is a smooth function. Let $(L(x))_{2\times 2}\in C^\infty$ be a matrix-valued function. 
 For any $x\in X$, $L(x)$ is positive definite and there exists c>0 such that 
$$c|\zeta|^2\leq \zeta L(x)\zeta, \ \ \forall \zeta\in\mathbb{R}^2, x\in X.$$

What I want to know is, whether $G(x,y)$ is continuously (smoothly) dependent of $\Omega$. In other words, let $G_{z}(x,y)$ be the Green's function of $-\Delta$ in $L(z)(\Omega):=\{L(z)x\mid x\in\Omega\}$. Is $G_z(x,y)$ continuous (smooth) about $z$?