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Journal of Convex Analysis 10 (2003), No. 1, 001--034
Copyright Heldermann Verlag 2003

Degenerate Perturbations of a Two-Phase Transition Model

Roberto Monti
Dip. di Matematica, UniversitÓ die Trento, Via Sommariva 14, 38050 Povo, Italy, rmonti@science.unitn.it

Francesco Serra Cassano
Dip. di Matematica, UniversitÓ die Trento, Via Sommariva 14, 38050 Povo, Italy, cassano@science.unitn.it

[Abstract-pdf]

We study the G-convergence as e approaches 0+ of the family of degenerate functionals Qe(u) = e Integral over W of <ADu, Du> dx + (1/e) Integral over W of W (u) dx, where A(x) is a symmetric, non negative n times n matrix on W (i.e. <A(x) ξ, ξ> ≥   0 for all x in W and x in Rn) with regular entries and W: R to [0, +infinity) is a double well potential having two isolated minimum points. Moreover, under suitable assumptions on the matrix A, we obtain a minimal interface criterion for the G-limit functional exploiting some tools of analysis in Carnot-Caratheodory spaces. We extend some previous results obtained for the non degenerate perturbations Qe in the classical gradient theory of phase transitions.

Keywords: Phase transitions, Γ-convergence, Carnot-Caratheodory spaces, minimal interface criterion.

MSC 2000: 49J45, 49Q05, 49Q20.

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