= Solution
Use the <method of continuity> with
$$
L_t=(1-t)\Delta+tL,
\qquad0\leq t\leq1.
$$
Every $L_t$ has the same uniform ellipticity and Hölder coefficient bounds, so the estimates in part (a) hold with one constant independent of $t$. They imply injectivity. The set of $t$ for which $L_t:C_0^{2,\alpha}(B)\to C^{0,\alpha}(B)$ is onto is open by the <bounded inverse theorem> and a Neumann-series perturbation, and closed by the uniform a priori estimate. It contains $0$ because the Dirichlet Laplacian is bijective. Connectedness of $[0,1]$ therefore gives surjectivity at $t=1$, and $L$ is bijective.
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