Interpolation inequalities in pattern formation

TL;DR

This paper establishes new interpolation inequalities relevant to pattern formation in physics, providing stronger estimates that improve understanding of energy bounds in phenomena like micromagnetics and superconductors.

Contribution

It introduces stronger interpolation inequalities with a novel geometric construction, extending previous weak estimates used in energy analysis of pattern formation.

Findings

Derived strong interpolation inequalities for pattern formation

Applied geometric construction method from superconductors study

Enhanced energy lower bounds in physical models

Abstract

We prove some interpolation inequalities which arise in the analysis of pattern formation in physics. They are the strong version of some already known estimates in weak form that are used to give a lower bound of the energy in many contexts (coarsening and branching in micromagnetics and superconductors). The main ingredient in the proof of our inequalities is a geometric construction which was first used by Choksi, Conti, Kohn, and one of the authors in the study of branching in superconductors.