Estimation of the continuity constants for Bogovski\uı and regularized Poincaré integral operators
Johnny Guzmán, Abner J. Salgado · arXiv (Cornell University) · 2020
We study the dependence of the continuity constants for the regularized Poincaré and Bogovski\uı integral operators acting on differential forms defined on a domain $Ω$ 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) $L^2(Ω,Λ^\ell)$ to $H^1(Ω,Λ^{\ell-1})$, $\ell \in \{1, \ldots, n\}$. For domains $Ω$ that are star shaped with respect to a ball $B$ we study the dependence of the constants on the ratio $diam(Ω)/diam(B)$. 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.