Weighted asymptotic Korn and interpolation Korn inequalities with singular weights

TL;DR

This paper establishes sharp weighted Korn and interpolation inequalities in thin domains with singular weights, with constants depending on the domain thickness in an optimal power-law manner, relevant for elasticity analysis.

Contribution

It introduces asymptotically sharp weighted Korn inequalities with optimal constants depending on domain thickness, extending classical results to singular weighted contexts.

Findings

Constants depend on thickness as $K=Ch^eta$ with optimal $eta$

Inequalities are sharp as $h o 0$

Applicable to Cartesian to polar coordinate transformations

Abstract

In this work we derive asymptotically sharp weighted Korn and Korn-like interpolation (or first and a half) inequalities in thin domains with singular weights. The constants $K$ (Korn's constant) in the inequalities depend on the domain thickness $h$ according to a power rule $K=Ch^\alpha,$ where $C>0$ and $\alpha\in R$ are constants independent of $h$ and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics $h^\alpha$ is optimal as $h\to 0.$ The choice of the weights is motivated by several factors, in particular a spacial case occurs when making Cartesian to polar change of variables in two dimensions.