Sharp angle estimates for second order divergence operators

TL;DR

This paper provides a comprehensive analysis of the sectoriality angles and numerical range estimates for second-order elliptic operators under minimal assumptions, including explicit bounds and applications across various Banach spaces.

Contribution

It offers new explicit estimates for sectoriality angles, extends results to operators with complex coefficients without geometric restrictions, and introduces a transfer principle for sectorial operators.

Findings

Explicit estimate for the numerical range sector angle.

Sectoriality angle bounds for elliptic operators on L^p spaces.

Transfer principle for Crouzeix-Delyon theorem with explicit constants.

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 $W^{1,2}(\Omega)$ 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 $L^p(\Omega)$ 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.