On a singular limit of the Kobayashi--Warren--Carter energy

TL;DR

This paper introduces a new topology called sliced graph convergence to analyze the Gamma limit of the Kobayashi-Warren-Carter energy, reducing the problem to a one-dimensional setting via slicing.

Contribution

It provides a novel representation formula for the Gamma limit of the energy in multi-dimensional domains using sliced graph convergence.

Findings

Representation formula for Gamma limit established

Sliced graph convergence is finer than L^1 convergence

Reduction to one-dimensional analysis via slicing

Abstract

By introducing a new topology, a representation formula of the Gamma limit of the Kobayashi-Warren-Carter energy is given in a multi-dimensional domain. A key step is to study the Gamma limit of a single-well Modica-Mortola functional. The convergence introduced here is called the sliced graph convergence, which is finer than conventional $L^1$ convergence, and the problem is reduced to a one-dimensional setting by a slicing argument.