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St. Petersburg Mathematical Journal

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A variational problem of phase transitions for a two-phase elastic medium with zero coefficient of surface tension


Author: V. G. Osmolovskiĭ
Translated by: N. B. Lebedinskaya
Original publication: Algebra i Analiz, tom 22 (2010), nomer 6.
Journal: St. Petersburg Math. J. 22 (2011), 1007-1022
MSC (2010): Primary 74B05
DOI: https://doi.org/10.1090/S1061-0022-2011-01181-1
Published electronically: August 22, 2011
MathSciNet review: 2760092
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Abstract: The variational problem on the equilibrium of a two-phase elastic medium is given in an extended form and is compared with the standard setting. The lower semicontinuity of the energy functional in the extended formulation is studied, and an example is constructed where no equilibrium states exist for a special class of residual strain tensors. In the case of isotropic media, a method is described for finding equilibrium states in explicit form. The notion of temperatures of phase transitions is introduced, their existence is proved, and their properties are studied.


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Additional Information

V. G. Osmolovskiĭ
Affiliation: Department of Mathematics and Mechanics, St. Petersburg State University, Universitetskaya Ul. 28, Stary Petergof, St. Petersburg 198504, Russia
Email: vicos@VO8167.spb.edu

DOI: https://doi.org/10.1090/S1061-0022-2011-01181-1
Keywords: Free surfaces, nonconvex variational problems, phase transitions in continuum mechanics
Received by editor(s): June 30, 2010
Published electronically: August 22, 2011
Additional Notes: Supported by RFBR (grant no. 08-01-00748)
Dedicated: Dedicated to Vasiliĭ Mikhaĭlovich Babich
Article copyright: © Copyright 2011 American Mathematical Society