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

TL;DR

This paper proves that certain higher-order elliptic systems on general bounded domains have $ ext{R}$-sectorial operators in $ ext{L}^p$ spaces, extending previous results to more general domains and boundary conditions.

Contribution

It generalizes the $ ext{L}^p$-extrapolation theorem of Shen to Banach space valued settings and arbitrary measurable sets, enabling the analysis on broader domains.

Findings

Proves $ ext{R}$-sectoriality of elliptic systems on general bounded domains.

Extends $ ext{L}^p$-extrapolation theorem to Banach space valued functions.

Establishes $ ext{R}$-sectoriality for a range of $p$ depending on domain dimension and order.

Abstract

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.