 Solution

Yes. The local $L^1$-to-$L^{\infty }$ estimate for solutions of a uniformly elliptic equation with bounded Hölder coefficients gives, whenever $B_r⋐B_R$,Thus $(u_k)$ is locally uniformly Cauchy and converges locally uniformly to a continuous representative of the $L^1$ limit $v$. Part (b) then applies, so $v\in C^2(B)$ and $Lv=0$.