Calculus of Variations and Geometric Measure Theory

G. Di Fratta - V. Slastikov - A. Zarnescu

Sufficient conditions for the existence of minimizing harmonic maps with axial symmetry in the small-average regime

created by difratta on 24 Jul 2023
modified on 19 Feb 2024


Submitted Paper

Inserted: 24 jul 2023
Last Updated: 19 feb 2024

Year: 2023


The paper concerns the analysis of global minimizers of a Dirichlet-type energy functional defined on the space of vector fields $H^1(S,T)$, where $S$ and $T$ are surfaces of revolution. The energy functional we consider is closely related to a reduced model in the variational theory of micromagnetism for the analysis of observable magnetization states in curved thin films. We show that axially symmetric minimizers always exist, and if the target surface $T$ is never flat, then any coexisting minimizer must have line symmetry. Thus, the minimization problem reduces to the computation of an optimal one-dimensional profile. We also provide a necessary and sufficient condition for energy minimizers to be axially symmetric.

Keywords: harmonic maps, Axial Symmetry