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