 Solution

Put $h=\log X$. The first equation becomeswhile the second remainsThe 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 $Y\in C_{\mathrm{loc}}^{1,\alpha }$. Hence the right side of the equation for $h$ is $C^{0,\alpha }$, and the Interior Schauder estimate gives $h$, and therefore $X$, in $C_{\mathrm{loc}}^{2,\alpha }$.

Expanding the $Y$ equation givesThe higher-order Schauder estimates now alternate between the equations for $h$ and $Y$, gaining derivatives at each step. Induction yields $(X,Y)\in C^{\infty }({\Omega }^{\prime };H)$ for every ${\Omega }^{\prime }⋐\Omega$.