Suppose I have a second order elliptic operator $L$, with smooth coefficients, defined on the plane $\mathbb{R}^2$. Consider the open unit disc $D$. Then $L$ is uniformly elliptic on $D$, by compactness argument. Now consider the Dirichlet problem, \begin{align*} Lu &= f,\\ u|_{\partial D} &= 0 \end{align*} where $f\in C^\infty_c(D)$ is compactly supported and smooth. This admits unique weak solution $u$, which is smooth by Elliptic regularity.
I would like to know, under what condition on $L$ can I extend $u$ smoothly, by setting it $0$ outside $D$.
Any help regarding this appreciated. Thanks!
EDIT:
(1) As discussed in the comments, simply extending the solution by zero on a larger disc and then appealing to uniqueness and regularity doesn't work. The obvious extension is a priori only continuous and not even a weak solution (on the larger disc).
(2) It is suggested here to apply Schauder existence theory repeatedly to obtain smooth solution up to boundary, but I am not sure if I understand the argument. Also does it help to extend beyond the boundary?