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arXiv · 1504.01677

On Poincar\'e, Friedrichs and Korns inequalities on domains and hypersurfaces

Duduchava Roland
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Abstract

The celebrated Poincar\'e and Friedrichs inequalities estimate the $\mathbb{L}_p$-norm of a function by the $\mathbb{L}_p$-norm of the gradient. We prove the Poincar\'e inequality for a domain $\Omega\subset \mathbb{R}^n$ and for a hypersurface $\mathcal{C}\subset\mathbb{R}^n$ based on open mapping theorem of Banach only. For a cylinder which has a hypersurface as a base, is prove stronger inequality, involving only the surface derivatives. Similar inequalities for the uniform $C$-norm are proved as well. We also estimate $\mathbb{H}^m_p$-norm of functions prove inequalities for some generalizations of the mentioned inequalities. We also prove Poincar\'e-Korns and Friedrichs-Korns inequalities for vector-func\-ti\-ons estimating the $\mathbb{L}_p$-norm of a function by the $\mathbb{L}_p$-norm of the deformation tensor only on domains and on hypersurfaces. The proofs are based on the paper \cite{Du10} of the author on Korns inequalities. And again, the norm of the function in a cylinder is estimated by is the deformation tensor on the base of the cylinder.

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Duduchava Roland. 2015-04-06. On Poincar\'e, Friedrichs and Korns inequalities on domains and hypersurfaces. https://arxiv.org/abs/1504.01677

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