Estimation of the continuity constants for Bogovski\u{\i} and regularized Poincar\'e integral operators

TL;DR

This paper investigates how the continuity constants of regularized Poincaré and Bogovski
2f integral operators depend on geometric properties of domains, providing estimates for higher order Sobolev norms and extensions to unions of star-shaped domains.

Contribution

It offers new estimates for the continuity constants of these operators based on domain geometry and extends results to unions of star-shaped domains.

Findings

Constants depend on the ratio of domain diameter to the inscribed ball's diameter.

Develops a method for estimating higher order Sobolev norms.

Extends results to certain unions of star-shaped domains.

Abstract

We study the dependence of the continuity constants for the regularized Poincar\'e and Bogovski\u{\i} integral operators acting on differential forms defined on a domain $\Omega$ of $\mathbb{R}^n$. We, in particular, study the dependence of such constants on certain geometric characteristics of the domain when these operators are considered as mappings from (a subset of) $L^2(\Omega,\Lambda^\ell)$ to $H^1(\Omega,\Lambda^{\ell-1})$, $\ell \in \{1, \ldots, n\}$. For domains $\Omega$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(\Omega)/diam(B)$. A program on how to develop estimates for higher order Sobolev norms is presented. The results are extended to certain classes of unions of star shaped domains.