Ginzburg-Landau equation and motion by mean curvature. I: Convergence
[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.
weak viscosity limit in the sense of Barles-Perthame
Ilmanen's generalization of Huisken's monotonicity formula
scientific article; zbMATH DE number 1326048
scientific article