The divDiv-complex and applications to biharmonic equations

ABSTRACT

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 $\mathrm{div}\mathrm{Div}$ 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 in Krendl et al. [A decomposition result for biharmonic problems and the Hellan–Herrmann–Johnson method. Electron Trans Numer Anal. 2016;45:257–282]. The discussion of the kernel of $\mathrm{div}\mathrm{Div}$ leads to (de Rham-like) closed and exact Hilbert complexes, the $\mathrm{div}\mathrm{Div}$-complex and its adjoint the $\mathrm{Grad}\mathrm{grad}$-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 ${\mathbb{R}}^N$.

KEYWORDS:

• Biharmonic equations
• Helmholtz decomposition
• Hilbert complexes

1991 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35G15
• 58A14

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 $\mathrm{\Gamma }$ is locally a graph of a Lipschitz function.

2 Note $\mathrm{Curl}\mathbf{M}=\mathrm{dev}\mathrm{Curl}\mathbf{M}$ for $\mathbf{M}\in {\mathcal{H}}_{\mathbb{S}}(\mathrm{Curl},\mathrm{\Omega })$ and thus for all $\mathbf{M}\in {\mathring{\mathcal{H}}}_{\mathbb{S}}(\mathrm{Curl},\mathrm{\Omega })\cap \mathrm{sym}\mathrm{Curl}{\mathcal{H}}_{\mathbb{T}}(\mathrm{sym}\mathrm{Curl},\mathrm{\Omega })$ ${|\mathbf{M}|}_{{\mathcal{L}}^2\left(\mathrm{\Omega }\right)}\le c_{\mathsf{R}}{|\mathrm{Curl}\mathbf{M}|}_{{\mathcal{L}}^2\left(\mathrm{\Omega }\right)}=c_{\mathsf{R}}{|\mathrm{dev}\mathrm{Curl}\mathbf{M}|}_{{\mathcal{L}}^2\left(\mathrm{\Omega }\right)}.$

Additional information

Funding

The research of the second author was supported by the Austrian Science Fund (FWF): project S11702-N23.