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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 5, Issue 4


Lusin's condition and the distributional determinant for deformations with finite energy

Duvan Henao
  • Laboratoire Jacques-Louis Lions, CNRS – Université Pierre et Marie Curie, Boîte Courrier 187, 75252 Paris Cedex 05, France
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/ Carlos Mora-Corral
Published Online: 2012-10-02 | DOI: https://doi.org/10.1515/acv.2011.016


Based on a previous work by the authors on the modelling of cavitation and fracture in nonlinear elasticity, we give an alternative proof of a recent result by Csörnyei, Hencl and Malý on the regularity of the inverse of homeomorphisms in the Sobolev space . With this aim, we show that the notion of fracture surface introduced by the authors in their model corresponds precisely to the original notion of cavity surface in the cavitation models of Müller and Spector (1995) and Conti and De Lellis (2003). We also find that the surface energy introduced in the model for cavitation and fracture is related to Lusin's condition (N) on the non-creation of matter.

A fundamental question underlying this paper is whether necessarily implies that the deformation opens no cavities. We show that this is not true unless Müller and Spector's condition INV for the non-interpenetration of matter is satisfied. Having thus provided an additional justification of its importance, we prove the stability of this condition with respect to weak convergence in the critical space . Combining this with the work by Conti and De Lellis, we obtain an existence theory for cavitation in this critical case.

Keywords: Elastic deformations; cavitation; surface energy; created surface; distributional determinant; Lusin's condition; non-interpenetration of matter

About the article

Received: 2009-12-01

Revised: 2011-05-03

Accepted: 2011-05-16

Published Online: 2012-10-02

Published in Print: 2012-10-01

Citation Information: , Volume 5, Issue 4, Pages 355–409, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv.2011.016.

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