Coupled singular perturbations for phase transitions

Article type: Research Article

Authors: Fonseca, Irene | Popovici, Cristina

Affiliations: Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA

Abstract: The \[$\varGamma(L^{1}(\varOmega;\mathbb{R}^{d}))$-limit of the sequence \[J_{\varepsilon}(u):=\frac{1}{\varepsilon}E_{\varepsilon}(u),\] where Eε is the family of anisotropic singular perturbations Eε(u):=∫Ωf(x,u(x),ε∇u(x)) dx of a non-convex functional of vector-valued functions E(u):=∫Ωf(x,u(x),∇u(x)) dx is obtained where f is a non-negative energy density satisfying f(x,u,0)=0 if and only if u∈{a,b}.

Keywords: [TeX:] \[$\varGamma$-convergence, phase transitions, singular perturbations, double-well potential

Journal: Asymptotic Analysis, vol. 44, no. 3-4, pp. 299-325, 2005