Asymptotic analysis of a second-order singular perturbation model for phase transitions

2009/11/04 by Marco Cicalese, Emanuele Nunzio Spadaro, Cicalese, Marco +4 · 1 citation
Computer Science · Engineering · Mathematics · #49J45 #49M25 #74B05 #76A15 #Advanced Mathematical Modeling in Engineering #Brake Systems and Friction Analysis #FOS: Mathematics #Functional Analysis (math.FA) #math.FA #msc:49J45 #msc:49M25 #msc:74B05 #msc:76A15

paper · pdf · doi:10.48550/arxiv.0911.0786

19 pages

arxiv created 2009/11/04 · openalex publication_date 2009/11/04 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the problem of the asymptotic description of a family of energies introduced by Coleman and Mizel in the theory of nonlinear second-order materials depending on an extra parameter k. By proving a new nonlinear interpolation inequality, we show that there exists a positive constant k₀ such that, for k<k₀, these energies Gamma-converge to a sharp interface functional. Moreover, for a special choice of the potential term in the energy, we provide an upper bound on the values of k such that minimizers cannot develop oscillations on some fine scale, thus improving previous estimates by Mizel, Peletier and Troy.

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