Journal of Mathematical Sciences

, Volume 154, Issue 1, pp 78–89 | Cite as

Dependence of the volume of an equilibrium phase on the temperature in the phase transition problem of continuum mechanics

  • V. G. Osmolovskii


We study a multi-valued function {ie078-01} associating with the temperature t the volume part of one of the phases of the equilibrium distribution in the phase transition problem of continuum mechanics. We establish the closedness of the graph of this function, the monotone decrease, and the localization of variability zones of its upper and lower envelopes. Bibliography: 6 titles.


Surface Tension Equilibrium State Equilibrium Distribution Isoperimetric Inequality Monotone Decrease 
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  1. 1.
    M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Moscow, Nauka, 1990.Google Scholar
  2. 2.
    V. G. Osmolovskii, “An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions” [in Russian], Sib. Mat. Zh. 35 (1994), no. 4, 835–846; English transl.: Sib. Math. J. 35 (1994), no. 4, 743–753.CrossRefMathSciNetGoogle Scholar
  3. 3.
    V. G. Osmolovskii, “Existence of equilibrium states in the one-dimensional phase transition problem” [in Russian], Vest. S. Peterburg. State Univ., Ser. 1 (2006), no. 3, 54–65.Google Scholar
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    L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, CRC Press, New York, 1992.MATHGoogle Scholar
  5. 5.
    V. G. Osmolovskii, “On the set of solutions to a variational phase transition problem of continuum mechanics” [in Russian], Probl. Mat. Anal. 35 (2007), 111–119; English transl.: J. Math. Sci., New York 144 (2007), no. 6, 4645–4654.Google Scholar
  6. 6.
    V. G. Osmolovskii, “Isoperimetric inequality and equilibrium states of a two-phase medium” [in Russian], Probl. Mat. Anal. 36 (2007), 81–88; English transl.: J. Math. Sci., New York 150 (2008), no. 1, 1877–1886.MathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2008

Authors and Affiliations

  1. 1.St. Petersburg State UniversitySt. PetersburgRussia

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