Korn inequality on irregular domains

Abstract

In this paper, we study the weighted Korn inequality on some irregular domains, e.g., s-John domains and domains satisfying quasihyperbolic boundary conditions. Examples regarding sharpness of the Korn inequality on these domains are presented. Moreover, we show that Korn inequalities imply certain Poincaré inequality.

Korn’s inequality and John domains

“…Even on simply connected planar domain, we do not know how to obtain the geometric counterpart of ( K p,w ). Indeed, as observed in [21], if two disjoint domains support ( K p,w ), respectively, then their union admits a ( K p,w ), possibly with larger constant. However, (K p ) or ( K p ) does not have this property.…”

“…Recently, there have been some studies concerning the Korn inequality for more irregular domains, such as Hölder domains and s-John domains with s > 1. It turns out the Korn inequality (K p ) does not hold for any 1 < p < ∞ on these domains, and instead there are weighted versions of the Korn inequality; see [2,3,21] for instance.…”

“…Recently, Acosta, Durán and Muschietti [1] proved the Korn inequality (in a different form than (K p )) holds for all p ∈ (1, ∞) on John domains (see Definition 1.3 below). We refer the reader to [2,3,7,8,11,21,29,31,37] for more studies on the Korn inequality. It is worth to mention that the Korn inequality (K p ) fails for p = 1 even on a cube; see the examples from [9,33].…”

On Korn's inequality at endpoints

“…, ε(u) := (ε ij (u)) 1 i,j n := ( 1 2 [3,13]. 近年来, 关于 Korn 不等式在一些不规则区域上的研究, 例如 Hölder 区域, s-John 区域 (s > 1), 见文献 [14][15][16].…”

Construction of a right inverse for the divergence in non-cylindrical time dependent domains

“…In addition, it is difficult to find a complete treatment of Hölder regular domains in the literature, even in the steady setting. Namely, first order Sobolev estimates in planar domains are treated in [3], extremely general results in [18] certainly cover the Hölder setting, and even sharp results are likely to follow from the treatise in [31], but it is still behind a modicum of work to extract the desired estimates from these references. The objective of the present paper is hence twofold.…”