论文标题
估计Bogovski \Uı的连续性常数和正规庞加莱集成运营商
Estimation of the continuity constants for Bogovski\uı and regularized Poincaré integral operators
论文作者
论文摘要
我们研究了常规Poincaré和Bogovski \uı的积分操作员的连续性常数的依赖性，这些操作员作用于$ \ Mathbb {r}^n $的域上定义的差异形式。特别是，当这些操作员被认为是（一个子集的）$ l^2（ω，λ^\ ell）$对$ h^1（ω，λ^{\λ^{\ ell-1}）$，$ h^1（ω，λ^\ ell）$，$ h^1（ω，λ^\ ell）$时，我们特别研究了此类常数对域的某些几何特征的依赖性。对于针对球$ b $的星形形状的域$ω$，我们研究常数对比率$ $ diam（ω）/直径（b）$的依赖性。提出了有关如何制定高阶Sobolev规范估计的计划。结果扩展到某些类别的星形域工会。
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.
