A finer singular limit of a single-well Modica–Mortola functional and its applications to the Kobayashi–Warren–Carter energy

An explicit representation of the Gamma limit a single-well Modica--Mortola functional is given for one-dimensional space under graph convergence which finer than conventional $L^1$-convergence or in measure. As an application, singular Kobayashi-Warren-Carter energy, popular materials science, given. Some compactness also established. Such formulas, as well compactness, useful to characterize minimizers energy. To convergence, new idea especially problem introduced. It change parameter variable by arc-length its graph, called unfolding this paper.