关于differential forms的Babuška-Aziz和Friedrichs-Velte不等式
Inequalities of Babuška-Aziz and Friedrichs-Velte for differential forms
摘要：
对于充分光滑的有界平面区域，Horgan和Payne在1983年证明了Babu{ \v s} ka-Aziz关于散度的右逆的不等式与Friedrichs关于共轭调和函数的不等式之间的等价性。在以前的文献[4]中，我们证明了这个等价性以及相关常数之间的相等性是真的，而不需要任何正则性条件。在三维空间中，Velte [9]研究了共轭调和函数概念的推广和相应的Friedrich不等式的推广，并证明了对于充分光滑的单连通域，散度和旋度的inf-sup条件是等价的。对于这种等价性，Zsupp{á}n [10]观察到我们的证明可以被修改，证明了相应常数之间的相等性，而无需对域进行正则性假设。在这里，我们制定了一个推广的弗里德里希不等式的共轭调和微分形式的有界开集在任何层面上，包含的情况下研究的霍根-佩恩和Velte作为特殊情况。我们还建立了相应的inf-sup条件或Babu{ \v s} ka-Aziz不等式，并证明了它们与Friedrichs不等式的等价性，包括相应常数之间的等价性.没有先验条件的规则性的开集，也没有对它的拓扑结构假设。
英文摘要：
For sufficiently smooth bounded plane domains, the equivalence between the inequalities of Babu{\v s}ka --Aziz for right inverses of the divergence and of Friedrichs on conjugate harmonic functions was shown by Horgan and Payne in 1983 [7]. In a previous paper [4] we proved that this equivalence, and the equality between the associated constants, is true without any regularity condition on the domain. In three dimensions, Velte [9] studied a generalization of the notion of conjugate harmonic functions and corresponding generalizations of Friedrich's inequality, and he showed for sufficiently smooth simply-connected domains the equivalence with inf-sup conditions for the divergence and for the curl. For this equivalence, Zsupp{á}n [10] observed that our proof can be adapted, proving the equality between the corresponding constants without regularity assumptions on the domain. Here we formulate a generalization of the Friedrichs inequality for conjugate harmonic differential forms on bounded open sets in any dimension that contains the situations studied by Horgan--Payne and Velte as special cases. We also formulate the corresponding inf-sup conditions or Babu{\v s}ka --Aziz inequalities and prove their equivalence with the Friedrichs inequalities, including equality between the corresponding constants. No a-priori conditions on the regularity of the open set nor on its topology are assumed.
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