Rectifiability of entropy defect measures in a micromagnetics model

TL;DR

This paper proves that the entropy defect measure for weak solutions of the eikonal equation in a micromagnetics context is concentrated on a rectifiable set, revealing the structure of discontinuities.

Contribution

It establishes the rectifiability of the entropy defect measure for solutions of the eikonal equation related to micromagnetics, linking measure concentration to jump discontinuities.

Findings

Entropy defect measure is concentrated on a 1-rectifiable set.

The rectifiable set detects jump discontinuities of solutions.

Provides geometric insight into the structure of solutions.

Abstract

We study the fine properties of a class of weak solutions $u$ of the eikonal equation arising as asymptotic domain of a family of energy functionals introduced in (Rivi\`ere T, Serfaty S. Limiting domain wall energy for a problem related to micromagnetics. Comm Pure Appl Math 2001; 54(3):294-338). In particular we prove that the entropy defect measure associated to $u$ is concentrated on a 1-rectifiable set, which detects the jump-type discontinuities of $u$.