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Singular perturbations of variational problems arising from a two-phase transition model

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Abstract

Given thatα, β are two Lipschitz continuous functions of Ω to ℝ+ and thatf(x, u, p) is a continuous function of\(\bar \Omega \) × ℝ × ℝN to [0, + ∞[ such that, for everyx, f(x,·, 0) reaches its minimum value 0 at exactly two pointsα(x) andβ(x), we prove the convergence ofFε(u) = (1/ε)∫Ωf (x, u, εDu) dx when the perturbation parameterε goes to zero. A formula is given for the limit functional and a general minimal interface criterium is deduced for a wide class of two-phase transition models. Earlier results of [19], [21], and [22] are extended with new proofs.

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  1. Mathématiques, Université de Toulon et du Var, Avenue de l'Université, BP132, 83957, La Garde Cedex, France

    Guy Bouchitte

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  1. Guy Bouchitte
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Communicated by R. Conti

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Bouchitte, G. Singular perturbations of variational problems arising from a two-phase transition model. Appl Math Optim 21, 289–314 (1990). https://doi.org/10.1007/BF01445167

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