Consider the annulus $\mathcal A:= B(0,2)\backslash B(0,1)$ in $\mathbb R^n$, $n\geq2$ and the divergence form elliptic operator $\operatorname{div}(A\nabla \cdot)$ on $\mathcal A$ where $A$ is a uniformly elliptic matrix with $L^\infty$ coefficients.

My question is: given any linear function $Lx$ with $|L|\leq1$ (in whatever reasonable matrix norm you want to consider), does there exist a continuous function $u$ on $\overline{\mathcal A}$ satisfying

1. $u|_{\partial B_1} = Lx$,
2. $u|_{\partial B_2} = 0$,
3. $|\operatorname{div}(A\nabla u)|(\overline{\mathcal A}) \leq C$ where the LHS is understood as a measure and the constant $C$ only depends on the ellipticity and $L^\infty$ bounds of the matrix $A$?

If $A$ is the identity (or very regular) the answer is easy as any $C^2$ extension will work, but I wonder what happens with more general operators...