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

Abstract

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

21 pages