Abstract
A variational problem on phase transitions in elastic media with nonhomogeneous boundary conditions is considered. Necessary conditions for a local minimum of the energy functional are established. These conditions are derived in the weak form of some integral identity, as well as in the form of the classical equilibrium equations. In the first case, no additional smoothness of the solution is required, whereas, in the second case, some additional conditions on the smoothness of the replacement field and the boundary of the interface of the phases are imposed. As was shown, even in the case of nonhomogeneous boundary conditions, the boundary of the interface of the phases intersects the boundary of the domain occupied by an elastic medium only at right angles. Bibliography: 3 titles.
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References
M. A. Grinfel0d, Methods of Continuum Mechanics in the Theory of Phase Transitions [in Russian], Nauka, Moscow (1990).
V. G. Osmolovskii, “An existence theorem and weak Lagrange equations for a variational problem of the theory of phase transitions, ” Sib. Math. J., 35, No. 4, 743-753 (1994).
V. G. Osmolovskii, Variational Problem on Phase Transitions in Mechanics of Continuous Media [in Russian], St.-Petersburg State University, St.-Petersburg (2000).
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Osmolovsksii, V.G. Necessary Conditions for the Extremum in a Variational Problem on Phase Transitions with Nonhomogeneous Boundary Conditions. Journal of Mathematical Sciences 106, 3015–3026 (2001). https://doi.org/10.1023/A:1011371605845
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DOI: https://doi.org/10.1023/A:1011371605845