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

Cagnetti, Filippo; Chambolle, Antonin; Scardia, Lucia (2022), Korn and Poincaré-Korn inequalities for functions with a small jump set, Mathematische Annalen, 383, p. 1179–1216. 10.1007/s00208-021-02210-w

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Type
Article accepté pour publication ou publié
Date
2022
Journal name
Mathematische Annalen
Volume
383
Publisher
Springer
Pages
1179–1216
Publication identifier
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Author(s)
Cagnetti, Filippo
Department of Mathematics [Brighton, Sussex]
Chambolle, Antonin
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Scardia, Lucia
School of Mathematical and Computer Sciences [MATHEMATICS DEPARTMENT OF HERIOT-WATT UNIVERSITY]
Abstract (EN)
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.
Subjects / Keywords
Elasticity; Fractures; Griffith energy; functions with bounded deformation; fine properties