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Singular perturbations of variational problems arising from a two-phase transition model. (English) Zbl 0695.49003

Summary: Given that \(\alpha\),\(\beta\) are two Lipschitz continuous functions of \(\Omega\) to \({\mathbb{R}}_+\) and that f(x,u,p) is a continuous function of \({\bar \Omega}\times {\mathbb{R}}\times {\mathbb{R}}^ N\) to \([0,+\infty [\) such that, for every x, f(x,\(\cdot,0)\) reaches its minimum value 0 at exactly two points \(\alpha\) (x) and \(\beta\) (x), we prove the convergence of \[ F^{\epsilon}(u)=(1/\epsilon)\int_{\Omega}f(x,u,\epsilon Du)dx \] when the perturbation parameter \(\epsilon\) 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 L. Modica [Arch. Ration. Mech. Anal. 98, 123-142 (1987; Zbl 0616.76004)] and N. Owen [“Nonconvex variational problems with general singular perturbations”, Preprint, Trans. Am. Math. Soc. (to appear)] and N. Owen and P. Sternberg [“Nonconvex problems with anisotropic perturbations”, Preprint (1988)] are extended with new proofs.

MSC:

49J10 Existence theories for free problems in two or more independent variables
49J45 Methods involving semicontinuity and convergence; relaxation
49K40 Sensitivity, stability, well-posedness

Citations:

Zbl 0616.76004
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References:

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