$\mathcal{R}$-sectoriality of higher-order elliptic systems on general bounded domains

On bounded domains $\mathrm{\Omega }\subset {\mathbb{R}}^d,d\ge 2$, reaching far beyond the scope of Lipschitz domains, we consider an elliptic system of order $2m$ in divergence form with complex ${\mathrm{L}}^{\mathrm{\infty }}$-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 $2m<d$, $p \in (\frac{2d}{d + 2 m} - \varepsilon , \frac{2d}{d - 2 m} + \varepsilon)$forsome$\varepsilon > 0$,andwhere$p \in (1 , \infty)$ in the case $2m\ge d$. 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.