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 (Korn's constant) in the inequalities depend on the domain thickness h according to a power rule K = Ch α , where C > 0 and α ∈ R are constants independent of h and the displacement field. The sharpness of the estimates is understood in the sense that the asymptotics h α is optimal as h → 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.

Asymptotics and Uniqueness of Solutions of the Elasticity System with the Mixed Dirichlet–Robin Boundary Conditions

“…We now claim that E a (u e , Ω) = Ω |x| a |ε(u e )| 2 dx = ∞ for e = 0. (26) First of all we show that if C 0 = 0 in (15), then…”

“…where σ(u e ) ≡ a ij kh ∂u ej ∂x h ν k . We claim that the constant C 0 is non-zero in (15). Indeed, if C 0 = 0, then |u e | ≤ C|x| 1−n and σ(u e )| ≤ C|x| −n .…”

“…We claim that E a (v, Ω) < ∞ for n − 2 ≤ a < n. Since C 0 = S e = S(S −1 C 0 ) = SS −1 C 0 = C 0 , it follows by (15) and 23that…”

“…In [14], the author extends the L 2 Korn interpolation inequality, as well as the second Korn inequalities, in thin domains, proved in [12], to the space L p for any 1 < p < ∞. Note the paper [15], in which the authors prove asymptotically sharp weighted Korn and Korn-type inequalities in thin domains with singular weights. The choice of weights is based on some factors; in particular, the spatial case arises when transforming Cartesian variables to polar change of variables in two dimensions.…”

“…where e = S −1 C 0 , and u e and u A are defined by (15) and (23) respectively. Obviously, v is a solution of (1) in Ω and…”