The uniform Korn - Poincar\'e inequality in thin domains

TL;DR

This paper establishes a uniform Korn-Poincaré inequality for thin shell domains around smooth hypersurfaces, showing the inequality's constant remains bounded as the shell thickness approaches zero, under certain conditions.

Contribution

It proves the uniform boundedness of the Korn-Poincaré constant in thin domains and identifies the optimality of the conditions related to Killing vector fields.

Findings

The Korn-Poincaré constant C_h remains bounded as h approaches 0.

The boundedness holds for vector fields in cones around the orthogonal complement of Killing field extensions.

The condition involving Killing fields is shown to be optimal.

Abstract

We study the Korn-Poincar\'e inequality: \|u\|_{W^{1,2}(S^h)} < C_h \|D(u)\|_{L^2(S^h)}, in domains S^h that are shells of small thickness of order h, around an arbitrary smooth and closed hypersurface S in R^n. By D(u) we denote the symmetric part of the gradient \nabla u, and we assume the tangential boundary conditions: u\vec n^h = 0 on \partial S^h. We prove that C_h remains uniformly bounded as h tends to 0, for vector fields u in any family of cones (with angle <\pi/2, uniform in h) around the orthogonal complement of extensions of Killing vector fields on S. We also show that this condition is optimal, as in turn every Killing field admits a family of extensions u^h, for which the ratio: \|u^h\|_{W^{1,2}(S^h)} / \|D(u^h)\|_{L^2(S^h)} blows up as h tends to 0, even if the domains S^h are not rotationally symmetric.