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 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.

Pointwise topological convergence and topological graph convergence of set-valued maps

Let X,Y be topological spaces and {Fn : n ? ?} be a sequence of set-valued maps from X to Y with the pointwise topological limit G and with the topological graph limit F. We give an answer to the question from ([19]): which conditions on X,Y and/or {F,G,Fn : n ? ?} are needed to F = G.