Let's consider a 2D-square with 4 euqal subsquares containing different dielectrics.
Inside the square domain, the unkown potential function $\Phi$ satisfies the Laplace equation:
$\nabla^2\Phi=0$
and satisfies the continuity conditions on the interfaces between differnet dielectrics:
$\Phi^+=\Phi^-$
$\frac{\partial\Phi^+}{\partial n}\epsilon^+=\frac{\partial\Phi^-}{\partial n}\epsilon^-$
If we apply Dirichlet Boundary conditions to the 4 edges of the square domain, then we can define a surface green function[1] G(x, y | $\eta$) to help solve the unknown function $\Phi$ inside the square domain:
$\Phi(x,y)=\oint_L\Phi(\eta)G(x,y|\eta)d\eta$
Coordinate (x, y) denotes a point inside the square domain while $\eta$ denotes the point on the domain boundary. The $\Phi(\eta)$ is the Dirichlete Boundary Condition and is known. L is the closed boundary of the square domain.
Assuming the length of the square domain's each edge is a, by numerical method, I found a property of G(x, y | $\eta$):
$$\frac{G(\frac{a}{2},\frac{a}{2}|\eta_i)}{\epsilon_i}=\frac{G(\frac{a}{2},\frac{a}{2}|\eta_j)}{\epsilon_j}$$
where the 2 points corresponding to $\eta_i$ and $\eta_j$ are centrosymmetric in reference to the central point of the square domain. $\epsilon_i$ and $\epsilon_j$ are the dielectric value of the subdomains containing them.
For special case, such as the dielectrics of each subsquare are the same, the analytic solution for G(x, y | $\eta$) does exist and the property holds, as done in [1] with variational separation method for Laplace equation. But I dont't know how to prove it for arbitary $\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4$ with analytic method. I don't know wether there is an analytic solution for G(x, y | $\eta$) with help of the centrosymmetry of the square domain. Or maybe we can only prove the property holds while the numerical solution converges to the true solution?
Thanks for your help!
Refs: [1] Y. Le Coz and R. B. Iverson, “A stochastic algorithm for high speed capacitance extraction in integrated circuits,” Solid State Electron., vol. 35, no. 7,pp.1005–1012, Jul. 1992.