Publications
2001
Issue 34

MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server end of 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

34/2001

A Gamma-convergence result for the two-gradient theory of phase transitions

Sergio Conti, Irene Fonseca and Giovanni Leoni

Abstract

The generalization to gradient vector fields of the classical double-well, singularly perturbed functionals, $I_{ϵ} (u; Ω) := \int_{Ω} \frac{1}{ϵ} W (\nabla u) + ϵ | \nabla^{2} u |^{2} d x$ where $W (ξ) = 0$ if and only if $ξ = A$ or $ξ = B$ , and $A - B$ is a rank-one matrix, is considered. Under suitable constitutive and growth hypotheses on W it is shown that $I_{ϵ} Γ$ -converge to $I (u; Ω) = {\begin{matrix} K \times H^{N - 1} (S (\nabla u) \cap Ω) & i f u \in W^{1, 1} (Ω; (R)^{d}), \nabla u \in B V (Ω; {A, B}), \\ + \infty & o t h e r w i s e, \end{matrix}$ where $K^{*}$ is the (constant) interfacial energy per unit area.

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Received:: 26.06.01

Published:: 26.06.01

MSC Codes:: 35G99, 35M99, 49J40, 49J45, 49K20, 74B20, 74G65, 74N99

Keywords:: phase transition, double-well potential, singular perturbations, gamma-convergence, vertical matching, lateral matching

Related publications

inJournal

2002 Repository Open Access

Sergio Conti, Irene Fonseca and Giovanni Leoni

A gamma-convergence result for the two-gradient theory of phase transitions

In: Communications on pure and applied mathematics, 55 (2002) 7, pp. 857-936

BibTex DOI: 10.1002/cpa.10035