Abstract
A variational problem describing phase transitions is considered. It is shown that a multi-vaued function associating with the set of parameters the set of solutions to the problem is continuous and has compact values. Bibliography: 5 titles.
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References
M. A. Grinfel’d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Moscow, Nauka, 1990.
V. G. Osmolovskii, “An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions” [in Russian], Sib. Mat. Zhurn. 35 (1994), no. 4, 835–846; English transl: Sib. Math. J., 35 (1994), no. 4, 743–753.
V. G. Osmolovskii, “Criterion for the lower semicontinuity of the energy functional of a two-phase elastic medium” [in Russian], Probl. Mat. Anal. 26 (2003), 215–254; English transl.: J. Math. Sci. New York 117 (2003), No. 3, 4211–4236.
L.C. Evans, R.F. Gariepi, Measure Theory and Fine Properties of Functions, CRC Press. Boca Raton, 1992
L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, Am. Math. Soc., 1990.
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Translated from Problemy Matematicheskogo Analiza, No. 35, 2007, pp. 111–119
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Osmolovskii, V.G. On the set of solutions to a variational phase transition problem of continuum mechanics. J Math Sci 144, 4645–4654 (2007). https://doi.org/10.1007/s10958-007-0300-5
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DOI: https://doi.org/10.1007/s10958-007-0300-5