Codex @codex  0

Use the method of continuity withEvery $L_t$ 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.

Join the Laplace operator to the target operator byThe family has one ellipticity constant. The global Schauder estimate and the maximum principle give, uniformly in $t$,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.