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

TL;DR

This paper extends Poincaré, Friedrichs, and Korn inequalities to domains and hypersurfaces, providing new proofs and stronger variants involving surface derivatives and deformation tensors, applicable in various norms.

Contribution

It introduces new proofs of classical inequalities on domains and hypersurfaces using Banach space theory, and establishes stronger inequalities involving surface derivatives and deformation tensors.

Findings

Proved Poincaré inequalities for domains and hypersurfaces using Banach space methods.

Established inequalities involving surface derivatives for cylindrical domains.

Derived Korn inequalities estimating functions via deformation tensors on domains and hypersurfaces.

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.