A Constructive Proof of Korn’s Scaled Inequalities for Shells

Cristinel Mardare; Thai Ha Nguyen

doi:10.4208/cmaa.2024-0007

A Constructive Proof of Korn’s Scaled Inequalities for Shells

Cristinel Mardare^*
, Thai Ha Nguyen

^*Corresponding author for this work

Department of Mathematics

Research output: Journal Publications and Reviews › RGC 21 - Publication in refereed journal › peer-review

Abstract

One of Korn’s scaled inequalities for shells asserts that the H¹-norm of a displacement field of a shell with thickness 2ε clamped on a portion of its lateral boundary, once scaled to a domain independent of ε, is bounded above by the L²-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order ε⁻¹ . We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190]. © 2025, Global Science Press. All rights reserved.

Original language	English
Pages (from-to)	87-111
Number of pages	25
Journal	Communications in Mathematical Analysis and Applications
Volume	4
Issue number	1
DOIs	https://doi.org/10.4208/cmaa.2024-0007
Publication status	Published - Feb 2025

Funding

The work described in this paper was substantially supported by a Grant from the City University of Hong Kong (Project No. 7005608).

Research Keywords

asymptotic analysis
Korn inequalities
shells

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title = "A Constructive Proof of Korn{\textquoteright}s Scaled Inequalities for Shells",

abstract = "One of Korn{\textquoteright}s scaled inequalities for shells asserts that the H1-norm of a displacement field of a shell with thickness 2ε clamped on a portion of its lateral boundary, once scaled to a domain independent of ε, is bounded above by the L2-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order ε−1 . We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190]. {\textcopyright} 2025, Global Science Press. All rights reserved.",

keywords = "asymptotic analysis, Korn inequalities, shells",

author = "Cristinel Mardare and Nguyen, \{Thai Ha\}",

year = "2025",

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doi = "10.4208/cmaa.2024-0007",

language = "English",

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TY - JOUR

T1 - A Constructive Proof of Korn’s Scaled Inequalities for Shells

AU - Mardare, Cristinel

AU - Nguyen, Thai Ha

PY - 2025/2

Y1 - 2025/2

N2 - One of Korn’s scaled inequalities for shells asserts that the H1-norm of a displacement field of a shell with thickness 2ε clamped on a portion of its lateral boundary, once scaled to a domain independent of ε, is bounded above by the L2-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order ε−1 . We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190]. © 2025, Global Science Press. All rights reserved.

AB - One of Korn’s scaled inequalities for shells asserts that the H1-norm of a displacement field of a shell with thickness 2ε clamped on a portion of its lateral boundary, once scaled to a domain independent of ε, is bounded above by the L2-norm of the corresponding scaled infinitesimal strain tensor field multiplied by a constant of order ε−1 . We give a constructive proof to this inequality, and to other two inequalities of this type, which is thus different from the original proof of Ciarlet et al. [Arch. Rational Mech. Anal. 136 (1996), 163–190]. © 2025, Global Science Press. All rights reserved.

KW - asymptotic analysis

KW - Korn inequalities

KW - shells

UR - https://www.scopus.com/pages/publications/105024906623

UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-105024906623&origin=recordpage

U2 - 10.4208/cmaa.2024-0007

DO - 10.4208/cmaa.2024-0007

M3 - RGC 21 - Publication in refereed journal

SN - 2790-1920

VL - 4

SP - 87

EP - 111

JO - Communications in Mathematical Analysis and Applications

JF - Communications in Mathematical Analysis and Applications

IS - 1

ER -

A Constructive Proof of Korn’s Scaled Inequalities for Shells

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