Weighted asymptotic Korn and interpolation Korn inequalities with singular weights

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 K (Korn’s constant) in the inequalities depend on the domain thickness h h according to a power rule K = C h α , K=Ch^\alpha , where C > 0 C>0 and α ∈ R \alpha \in R are constants independent of h h and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics h α h^\alpha is optimal as h → 0. h\to 0. The choice of the weights is motivated by several factors; in particular a spatial case occurs when making Cartesian to polar change of variables in two dimensions.