设 $f(u, v)$ 具有二阶连续偏导数，且满足 $\frac{\partial^2 f}{\partial u^2}+\frac{\partial^2 f}{\partial v^2}=1$. 又

$$
g(x, y)=f\left[x y, \frac{1}{2}\left(x^2-y^2\right)\right]
$$


若 $G(x, y)=\frac{\partial^2 g}{\partial x^2}+\frac{\partial^2 g}{\partial y^2}$ ，试求 $G(x, y)$ 的表达式.