= Solution
Put $h=\log X$. The first equation becomes
$$
\Delta h=\frac{\Delta X}{X}-\frac{|\nabla X|^2}{X^2}
=-\frac{|\nabla Y|^2}{X^2},
$$
while the second remains
$$
\operatorname{div}(X^{-2}\nabla Y)=0.
$$
The assumed $C^{0,\alpha}$ regularity and the bounds away from zero make $X^{-2}$ a $C^{0,\alpha}$ uniformly elliptic coefficient. The divergence-form $C^{1,\alpha}$ estimate from part (d) first gives $Y\in C^{1,\alpha}_{\mathrm{loc}}$. Hence the right side of the equation for $h$ is $C^{0,\alpha}$, and the <Interior Schauder estimate> gives $h$, and therefore $X$, in $C^{2,\alpha}_{\mathrm{loc}}$.
Expanding the $Y$ equation gives
$$
\Delta Y-2\nabla h\cdot\nabla Y=0.
$$
The higher-order Schauder estimates now alternate between the equations for $h$ and $Y$, gaining derivatives at each step. Induction yields $(X,Y)\in C^\infty(\Omega';H)$ for every $\Omega'\Subset\Omega$.
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