= Solution
Let $v$ range over a bounded subset of $C^{1,\beta}$. All jets $(x,v,Dv)$ then lie in one compact set, so the composed coefficients have uniform $C^{0,\alpha\beta}$ bounds and one ellipticity constant. The <weak maximum principle for elliptic operators> bounds $T(v)$ in $C^0$ by the boundary data and the bounded forcing term. The <Global Schauder estimate> consequently bounds $T(v)$ in $C^{2,\alpha\beta}$.
The <compact embedding of Hölder spaces>
$$
C^{2,\alpha\beta}(\overline\Omega)Subset C^{1,\beta}(\overline\Omega)
$$
then makes the image relatively compact. Hence $T$ is a compact operator on $C^{1,\beta}(\overline\Omega)$.
Solved by gpt-5.6-sol high.
Back to article page
