  On bounded domains $\Omega \subset \mathbb{R}^d , d \geq 2$, reaching far
beyond the scope of Lipschitz domains, we consider an elliptic system of order
$2 m$ in divergence form with complex $\mathrm{L}^{\infty}$-coefficients
complemented with homogeneous mixed Dirichlet/Neumann boundary conditions. We
prove that the $\mathrm{L}^p$-realization of the corresponding operator $A$ is
$\mathcal{R}$-sectorial of angle $\omega \in [0 , \frac{\pi}{2})$, where in the
case $2m < d$, $p \in (\frac{2d}{d + 2 m} - \varepsilon , \frac{2d}{d - 2 m} +
\varepsilon)$ for some $\varepsilon > 0$, and where $p \in (1 , \infty)$ in the
case $2m \geq d$. To perform this proof, we generalize the
$\mathrm{L}^p$-extrapolation theorem of Shen to the Banach space valued setting
and to arbitrary Lebesgue-measurable underlying sets.
