FALL 2021 Nonlinear Analysis Seminar Series

★DISTINGUISHED LECTURE

Professor Yoshikazu Giga,  The University of Tokyo

Title: On a singular limit of a single-well Modica-Mortola functional and its applications

Abstract:

It is important to describe the motion of phase boundaries by macroscopic energy in the process of phase transitions. Typical energy describing the phenomena is the van der Waals energy, which is also called a Modica-Mortola functional with a double-well potential or the Allen-Cahn functional. It turns out that it is also important to consider the Modica-Mortola functional with a single-well potential since it is often used in various settings including the Kobayashi-Warren-Carter energy, which is popular in materials science. It is very fundamental to understand the singular limit of such a type of energies as the thickness parameter of a diffuse interface tends to zero. In the case of double-well potentials, such a problem is well-studied and it is formulated, for example, as the Gamma limit under L1

convergence.

However, if one considers the Modica-Mortola functional, it turns out that L1

convergence is too rough even in the one-dimensional problem.

We characterize the Gamma limit of a single-well Modica-Mortola functional under the topology which is finer than L1

topology. In a one-dimensional case, we take the graph convergence. In higher-dimensional cases, it is more involved. As an application, we give an explicit representation of a singular limit of the Kobayashi-Warren-Carter energy. Since the higher-dimensional cases can be reduced to the one-dimensional case by a slicing argument, studying the one-dimensional case is very fundamental. A key idea to study the one-dimensional case is to introduce “an unfolding of a function” by changing an independent variable by the arc-length parameter of its graph. This is based on a joint work with Jun Okamoto (The University of Tokyo), Masaaki Uesaka (The University of Tokyo, Arithmer Inc.), and Koya Sakakibara (Okayama University of Science, RIKEN).

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