Korn and Poincaré-Korn inequalities for functions with a small jump set

Filippo Cagnetti; Antonin Chambolle; Lucia Scardia

doi:10.1007/s00208-021-02210-w

Korn and Poincaré-Korn inequalities for functions with a small jump set

Filippo Cagnetti
, Antonin Chambolle
, Lucia Scardia

Research output: Contribution to journal › Article › peer-review

19 Citations (Scopus)

63 Downloads (Pure)

Abstract

In this paper we prove a regularity and rigidity result for displacements in GSBD^p, for every p>1 and any dimension n≥2. We show that a displacement in GSBD^p with a small jump set coincides with a W^1,p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBD^p.

Original language	English
Pages (from-to)	1179–1216
Number of pages	38
Journal	Mathematische Annalen
Volume	383
Early online date	8 Jun 2021
DOIs	https://doi.org/10.1007/s00208-021-02210-w
Publication status	Published - Aug 2022

ASJC Scopus subject areas

General Mathematics

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title = "Korn and Poincar{\'e}-Korn inequalities for functions with a small jump set",

abstract = "In this paper we prove a regularity and rigidity result for displacements in GSBDp, for every p>1 and any dimension n≥2. We show that a displacement in GSBDp with a small jump set coincides with a W1,p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincar{\'e}-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBDp.",

author = "Filippo Cagnetti and Antonin Chambolle and Lucia Scardia",

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T1 - Korn and Poincaré-Korn inequalities for functions with a small jump set

AU - Cagnetti, Filippo

AU - Chambolle, Antonin

AU - Scardia, Lucia

N1 - Funding Information: The authors would like to thank the referee for carefully reading the manuscript and for providing several helpful suggestions. The authors acknowledge the hospitality and support of the INI under the Grant EP/R014604/1. Publisher Copyright: © 2021, The Author(s).

PY - 2022/8

Y1 - 2022/8

N2 - In this paper we prove a regularity and rigidity result for displacements in GSBDp, for every p>1 and any dimension n≥2. We show that a displacement in GSBDp with a small jump set coincides with a W1,p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBDp.

AB - In this paper we prove a regularity and rigidity result for displacements in GSBDp, for every p>1 and any dimension n≥2. We show that a displacement in GSBDp with a small jump set coincides with a W1,p function, up to a small set whose perimeter and volume are controlled by the size of the jump. This generalises to higher dimension a result of Conti, Focardi and Iurlano. A consequence of this is that such displacements satisfy, up to a small set, Poincaré-Korn and Korn inequalities. As an application, we deduce an approximation result which implies the existence of the approximate gradient for displacements in GSBDp.

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DO - 10.1007/s00208-021-02210-w

M3 - Article

SN - 0025-5831

VL - 383

SP - 1179

EP - 1216

JO - Mathematische Annalen

JF - Mathematische Annalen

ER -

Korn and Poincaré-Korn inequalities for functions with a small jump set

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