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

Abstract

On bounded domains \(\varOmega \subset {\mathbb {R}}^d , d \ge 2\), 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 \(2m \ge d\), \(p \in (1,\infty )\), and where \(p \in (\frac{2d}{d + 2 m} - \varepsilon , \frac{2d}{d - 2 m} + \varepsilon )\) for some \(\varepsilon > 0\) in the case \(2m < d\). 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.