We obtain sufficient conditions for the noncoincidence of the phase transition temperatures and illustrate this result by examples of problems in two-phase elastic media. We also indicate some cases where equilibrium states exist or not depending on the values of the temperature lying between the lower and upper phase transition temperatures. Bibliography: 10 titles.
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M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Nauka, Moscow (1990).
V. G. Osmolovskii, “Criterion for the lower semicontinuity of the energy functional of a two-phase elastic medium” [in Russian], Probl. Mat. Anal. 26, 215–254 (2003); English transl.: J. Math. Sci. (New York) 117, No. 3, 4211–4236 (2003).
B. Dacorogna, Direct Methods in the Calculus of Variations, Berlin (1989).
B. Dacorogna, G. Pisante, and A. Ribero, “On non quasiconvex problem of the calculus of variations,” Sect. Math. EPFL, 1015 Lausanne, Switzerland. February 4, 2005.
V. G. Osmolovskii, “Exact solutions to the variational problem of the phase transition theory in continuum mechanics” [in Russian], Probl. Mat. Anal. 27, 171–206 (2004); English transl.: J. Math. Sci. (New York) 120, No. 2, 1167–1190 (2004).
V. G. Osmolovskii, “Existence of equilibrium states in the one-dimensional phase transition problem” [in Russian], Vest. S. Peterburg. State Univ. Ser. 1, No. 3, 54–65 (2006).
V. G. Osmolovskii, “On the solvability of a variational problem about phase transitions in continuum mechanics” [in Russian], Probl. Mat. Anal. 38 61–72 (2008); English transl.: J. Math. Sci. (New York) 156, No. 4, no. 2, 632–643 (2009).
V. G. Osmolovskii, “Existence of phase transition temperatures of a nonhomogeneous anisotropic two-phase elastic medium” [in Russian], Probl. Mat. Anal. 31 59-66 (2005); English transl.: J. Math. Sci. (New York) 132, No. 4, 441–450 (2006).
V. G. Osmolovskii, “Dependence of the temperature of phase transitions on the size of the domain” [in Russian], Zap. Nauchn. Semin. POMI 310, 98–114 (2004); English transl.: J. Math. Sci. (New York) 132, No. 3, 304–312 (2006).
S. Muller, “Variational models for microstructure and phase transitions,” Preprint. Max-Planck-Institut, Leipzig, Lecture notes N2, (1998).
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Dedicated to Nina Nikolaevna Uraltseva
Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 37–48.
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Osmolovskii, V.G. On the phase transition temperature in a variational problem of elasticity theory for two-phase media. J Math Sci 159, 168–179 (2009). https://doi.org/10.1007/s10958-009-9433-z
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DOI: https://doi.org/10.1007/s10958-009-9433-z