= Solution
Join the <Laplace operator> to the target operator by
$$
L_tu=((1-t)\delta_{ij}+ta_{ij})\partial_{ij}u,
\qquad0\leq t\leq1.
$$
The family has one ellipticity constant. The global <Schauder estimate> and the maximum principle give, uniformly in $t$,
$$
\lVert u\rVert_{C^{2,\alpha}(\overline\Omega)}
\leq C\lVert L_tu\rVert_{C^\alpha(\overline\Omega)}
$$
for zero boundary data. Let $I$ contain those $t$ for which $L_t:C_0^{2,\alpha}\to C^\alpha$ is onto. The assumed Laplace solvability gives $0\in I$; the <bounded inverse theorem> and small perturbations make $I$ open; and the uniform estimate plus compactness of lower Hölder embeddings makes $I$ closed. The <method of continuity> yields $I=[0,1]$. At $t=1$ this gives the required solution, and the maximum principle gives uniqueness.
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