Ginzburg-Landau equation and motion by mean curvature. I: Convergence
type: Item
instance of: scholarly article
title: Ginzburg-Landau equation and motion by mean curvature. I: Convergence
zbMATH Open document ID: 0935.35060
DOI: 10.1007/BF02921628
author: H. Mete Soner
published in: The Journal of Geometric Analysis
publication date: 2000-04-24T00:00:00Z
review text: [For part II, see ibid., 477-491 (1997; Zbl 0935.35063).] The paper deals with the asymptotic study as \(\varepsilon\rightarrow 0\) of solutions to the Ginzburg-Landau equation \(u_t^\varepsilon - \Delta u^\varepsilon + 2/\varepsilon^2 u^\varepsilon (u^\varepsilon -1) (u^\varepsilon +1)=0\) in \({\mathbb R}^d\times [0,+\infty)\). The main result establishes the existence of two open, disjoint subsets \({\mathcal P}\), \({\mathcal N}\) of \({\mathbb R}^d\times (0,+\infty)\) and of a subsequence \(\varepsilon_n \rightarrow 0\) such that: (i) \(u^{\varepsilon_n}\rightarrow 1\), uniformly on bounded subsets of \({\mathcal P}\); (ii) \(u^{\varepsilon_n}\rightarrow -1\), uniformly on bounded subsets of \({\mathcal N}\); (iii) \(\Gamma\) = complement of the set \({\mathcal P}\cup{\mathcal N}\) has Hausdorff dimension \(d\). The main theoretical instruments of the analysis developed in the paper are viscosity solutions of parabolic equations, weak viscosity limit in the sense of Barles-Perthame, weak solutions of mean curvature flow and Ilmanen's generalization of Huisken's monotonicity formula.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1326048
zbMATH Keywords: weak viscosity limit in the sense of Barles-Perthame
zbMATH Keywords: Ilmanen's generalization of Huisken's monotonicity formula
zbMATH Keywords: phase transition
zbMATH Keywords: viscosity solution
reviewed by: Vicenţiu D. Rădulescu
MaRDI profile type: Publication
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