Sharp angle estimates for second order divergence operators

TL;DR

The article develops sharp angle estimates for the numerical range of second-order divergence-form elliptic operators with mixed boundary conditions, formulated via Dirichlet forms on closed subspaces of $W^{1,2}(\Omega)$. It first derives a sharp, data-driven inclusion $N(\mathfrak{t}) \subset \Sigma_\alpha$ for the form, then translates this to sectoriality and analytic semigroup properties for the operator on $L^2(\Omega)$ and its $L^p(\Omega)$ realizations, including both real and complex coefficients. A key advancement is the extension of Crouzeix–Delyon type bounds to unbounded sectorial operators and explicit $\mathrm{H}^\infty$-calculus angles across $L^p$ spaces, with refined $p$-ellipticity-based estimates for complex coefficients. The work further transports these sectorial properties to negative Sobolev scales via square-root techniques, establishing maximal parabolic regularity on $W^{-1,q}_D(\Omega)$ and, by interpolation, on negative Bessel-type scales. Overall, the paper provides a streamlined, geometry-light framework linking numerical range, sectoriality, functional calculus, and regularity across a hierarchy of function spaces, with explicit constants tied to problem data.

Abstract

This article is about the (minimal) sector containing the numerical range of the principal part of a linear second-order elliptic differential operator defined by a form on closed subspaces V of the first-order Sobolev space $W^{1,2}(Ω)$ incorporating mixed boundary conditions. We collect a comprehensive array of results on the angle of sectoriality and the $H^\infty$-angle attached to realizations of the elliptic operator. We thereby consider the operator in several scales of Banach spaces: the Lebesgue space, the negative Sobolev space, and their interpolation scale. For the latter two types of spaces, we rely on recent results regarding the Kato square root property. We focus on minimal assumptions on geometry, and we consider both real and complex coefficients. Not all results presented are new, but we strive for a streamlined and comprehensive overall picture from several branches of operator theory, and we complement the existing results with several new ones, in particular aiming at explicit estimates built on readily accessible problem data. This concerns for example a new estimate on the angle of the sector containing the numerical range of a linear, continuous and coercive Hilbert space operator, but also an explicit estimate for the angle of sectoriality for the elliptic operator on $L^p(Ω)$ with complex coefficients without any assumptions on geometry and a general transfer principle for the Crouzeix-Delyon theorem from bounded operators to sectorial ones, keeping the explicit constant.

Theorems & Definitions (59)

• Lemma 2.1
• proof
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Theorem 2.5
• proof
• Remark 2.6
• Corollary 2.7
• Definition 3.2: Sectorial operator