Existence and uniqueness of domain walls for notched ferromagnetic nanowires

¹ IRMA, Université de Strasbourg, CNRS UMR 7501, Inria, 7 rue René Descartes, 67084 Strasbourg, France
² Univ. Lille, Inria, CNRS, UMR 8524 – Laboratoire Paul Painlevé, 59000 Lille, France
³ Univ. Bordeaux, CNRS, Bordeaux INP, IMB, UMR 5251, F-33400 Talence, France
⁴ Institut Élie Cartan de Lorraine, École des Mines de Nancy, Boulevard des Aiguillettes, 54506 Vandoeuvre-lès-Nancy, France
⁵ Institut d’études avancées de l’université de Strasbourg (USIAS), France
⁶ Institut Universitaire de France (IUF), France

^* Corresponding author: ludovic.godard-cadillac@math.u-bordeaux.fr

Received: 21 January 2025
Accepted: 5 June 2025

Abstract

In this article, we investigate a simple model of notched ferromagnetic nanowires using tools from calculus of variations and critical point theory. Specifically, we focus on the case of a single unimodal notch and establish the existence and uniqueness of the critical point of the energy. This is achieved through a lifting argument, which reduces the problem to a generalized Sturm-Liouville equation. Uniqueness is demonstrated via a Mountain-Pass argument, where the assumption of two distinct critical points leads to a contradiction. Additionally, we show that the solution corresponds to a system of magnetic spins characterized by a single domain wall localized in the vicinity of the notch. We further analyze the asymptotic decay of the solution at infinity and explore the symmetric case using rearrangement techniques.