Rotationally symmetric p-harmonic maps from D^2 to S^2

TL;DR

This paper studies rotationally symmetric p-harmonic maps from the unit disk to the sphere, proving uniqueness of the energy minimizer and characterizing infinitely many solutions to the Euler-Lagrange equation.

Contribution

It establishes the existence, uniqueness, and smoothness of the energy minimizer, and fully characterizes all solutions to the Euler-Lagrange equation for these maps.

Findings

Unique smooth energy minimizer exists.

Infinitely many solutions to the Euler-Lagrange equation.

Complete characterization of all solutions.

Abstract

We consider rotationally symmetric $p$-harmonic maps from the unit disk $D^2\subset\real^2$ to the unit sphere $S^2\subset\real^3$, subject to Dirichlet boundary conditions and with $1<p<\infty$. We show that the associated energy functional admits a unique minimizer which is of class $C^{\infty}$ in the interior and $C^1$ up to the boundary. We also show that there exist infinitely many global solutions to the associated Euler-Lagrange equation and we completely characterize them.