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
  It is shown that the first biharmonic boundary value problem on a topologically trivial domain in 3D is equivalent to three (consecutively to solve) second-order problems. This decomposition result is based on a Helmholtz-like decomposition of an involved non-standard Sobolev space of tensor fields and a proper characterization of the operator div-Div acting on this space. Similar results for biharmonic problems in 2D and their impact on the construction and analysis of finite element methods have been recently published by the second author. The discussion of the kernel of div-Div leads to (de Rham-like) closed and exact Hilbert complexes, the div-Div-complex and its adjoint the Grad-grad-complex, involving spaces of trace-free and symmetric tensor fields. For these tensor fields we show Helmholtz type decompositions and, most importantly, new compact embedding results. Almost all our results hold and are formulated for general bounded strong Lipschitz domains of arbitrary topology. There is no reasonable doubt that our results extend to strong Lipschitz domains in arbitrary dimensions. 
Dirk Pauly and Walter Zulehner
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