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

Published electronically:
August 22, 2011

MathSciNet review:
2760092

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Abstract | References | Similar Articles | Additional Information

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.

**1.**M. A. Grinfel′d,*Metody mekhaniki sploshnykh sred v teorii fazovykh prevrashchenii*, “Nauka”, Moscow, 1990 (Russian). With an English summary. MR**1128092****2.**Lawrence C. Evans and Ronald F. Gariepy,*Measure theory and fine properties of functions*, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR**1158660****3.**V. G. Osmolovskiĭ,*An existence theorem and the weak form of Lagrange equations for a variational problem in the theory of phase transformations*, Sibirsk. Mat. Zh.**35**(1994), no. 4, 835–846, iii (Russian, with Russian summary); English transl., Siberian Math. J.**35**(1994), no. 4, 743–753. MR**1302437**, 10.1007/BF02106618**4.**V. G. Osmolovskij,*Phase transition in the mechanics of continuous media for big loading*, Math. Nachr.**177**(1996), 233–250. MR**1374951**, 10.1002/mana.19961770113**5.**Lawrence C. Evans,*Weak convergence methods for nonlinear partial differential equations*, CBMS Regional Conference Series in Mathematics, vol. 74, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1990. MR**1034481****6.**V. G. Osmolovskii,*Criterion for the lower semicontinuity of the energy functional of a two-phase elastic medium*, J. Math. Sci. (N. Y.)**117**(2003), no. 3, 4211–4236. Nonlinear problems and function theory. MR**2027456**, 10.1023/A:1024820721057**7.**Stefan Müller,*Microstructures, phase transitions and geometry*, European Congress of Mathematics, Vol. II (Budapest, 1996) Progr. Math., vol. 169, Birkhäuser, Basel, 1998, pp. 92–115. MR**1645820****8.**V. G. Osmolovskii,*On the phase transition temperature in a variational problem of elasticity theory for two-phase media*, J. Math. Sci. (N. Y.)**159**(2009), no. 2, 168–179. Problems in mathematical analysis. No. 41. MR**2544035**, 10.1007/s10958-009-9433-z**9.**V. G. Osmolovskii,*Exact solutions to the variational problem of the phase transition theory in continuum mechanics*, J. Math. Sci. (N. Y.)**120**(2004), no. 2, 1167–1190. Applications of mathematical analysis. MR**2099066**, 10.1023/B:JOTH.0000014845.60594.5f**10.**V. G. Osmolovskii,*Existence of phase transitions temperatures of a nonhomogeneous anisotropic two-phase elastic medium*, J. Math. Sci. (N. Y.)**132**(2006), no. 4, 441–450. Problems in mathematical analysis. No. 31. MR**2197338**, 10.1007/s10958-005-0510-7

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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