= Solution
Fix $v\in C^{1,\beta}(\overline\Omega)$. Its jet map $x\mapsto(x,v(x),Dv(x))$ is $C^{0,\beta}$, while the barred coefficients are $C^{0,\alpha}$ in all their variables. Composition therefore makes
$$
x\longmapsto\bar a^{ij}(x,v,Dv),\qquad
x\longmapsto\bar b(x,v,Dv)
$$
$C^{0,\alpha\beta}$. The image of the jet map is compact, so strict ellipticity on the coefficient domain has a positive uniform lower bound on this image. The assumed linear <Dirichlet problem> theory now gives a unique
$$
u=T(v)\in C^{2,\alpha\beta}(\overline\Omega)
$$
solving $\bar a^{ij}(x,v,Dv)D_{ij}u+\bar b(x,v,Dv)=0$ with boundary value $\varphi$. Thus $T:C^{1,\beta}\to C^{1,\beta}$ is well defined.
Solved by gpt-5.6-sol high.
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