Abstract
Abstract An explicit representation of the Gamma limit of a single-well Modica–Mortola functional is given for one-dimensional space under the graph convergence which is finer than conventional L 1 {L^{1}} -convergence or convergence in measure. As an application, an explicit representation of a singular limit of the Kobayashi–Warren–Carter energy, which is popular in materials science, is given. Some compactness under the graph convergence is also established. Such formulas as well as compactness are useful to characterize the limit of minimizers of the Kobayashi–Warren–Carter energy. To characterize the Gamma limit under the graph convergence, a new idea which is especially useful for one-dimensional problems is introduced. It is a change of parameter of the variable by arc-length parameter of its graph, which is called unfolding by the arc-length parameter in this paper.
Highlights
We are interested in a singular limit called the Gamma limit of a single-well Modica–Mortola functional under the graph convergence, the convergence with respect to the Hausdorff distance of graphs, which is finer than conventional L1 -convergence or convergence in measure
A single-well Modica–Mortola functional is introduced by Ambrosio and Tortorelli [2, 3] to approximate the Mumford–Shah functional [26]
Later in [23, 33], it was extended to multi-dimensional setting and the limit is a constant multiple of the surface area of the transition interface. This type of the Gamma convergence results as well as compactness is important to establish the convergence of local minimizer
Summary
We are interested in a singular limit called the Gamma limit of a single-well Modica–Mortola functional under the graph convergence, the convergence with respect to the Hausdorff distance of graphs, which is finer than conventional L1 -convergence or convergence in measure. Later in [23, 33], it was extended to multi-dimensional setting and the limit is a constant multiple of the surface area of the transition interface This type of the Gamma convergence results as well as compactness is important to establish the convergence of local minimizer ([21]) as well as the global minimizer. For the Steiner problem, such approximation as proposed ([22]) and its Gamma limit is established ([4]) All these problems the problem is closer to the Ambrosio–Tortorelli inhomogenization of the Dirichlet energy, not of the total variation. The authors are grateful to Professor Ken Shirakawa for letting us know his recent results before publication as well as development of researches on gradient flows of Kobayashi–Warren–Carter type energies
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