We show that the elasticity Hilbert complex with mixed boundary conditions on bounded strong Lipschitz domains is closed and compact. The crucial results are compact embeddings which follow by abstract arguments using functional analysis together with particular regular decompositions. Higher Sobolev order results are proved as well. This paper extends recent results on the de Rham Hilbert complex with mixed boundary conditions from [11] and recent results on the elasticity Hilbert complex with empty or full boundary conditions from [15].
研究了抽象Hilbert复形的迹算子和迹空间的新概念。我们引入迹空间作为商空间/零化子。我们刻画了相关迹算子的核和像，并讨论了迹空间之间的对偶关系。我们阐述了有界Lipschitz域上的Euclidean de Rham复形的经典边界迹的许多性质都植根于Hilbert复形的一般结构。我们到达抽象的跟踪希尔伯特复形，可以制定使用商空间/零化子。我们表明，如果一个希尔伯特复形承认稳定的“正规分解”与紧提升算子，那么相关的跟踪希尔伯特复形是Fredholm。在整个过程中，我们将讨论在三维欧几里得空间中的德拉姆复形的具体情况下抽象概念和结果的体现。
A Global div-curl-Lemma for Mixed Boundary Conditions in Weak Lipschitz Domains and a Corresponding Generalized $\mathrm{A}_{0}^{*}$-$\mathrm{A}_{1}$-Lemma in Hilbert Spaces
On Closed and Exact Grad-grad- and div-Div-Complexes, Corresponding Compact Embeddings for Tensor Rotations, and a Related Decomposition Result for Biharmonic Problems in 3D
Universität Duisburg-Essen（杜伊斯堡-埃森大学）；Johannes Kepler Universität Linz（约翰·开普勒林茨大学）
