Radial solutions of a fourth order Hamiltonian stationary equation

TL;DR

This paper investigates radial solutions to a fourth order Hamiltonian stationary equation, revealing that in two dimensions solutions are necessarily special Lagrangian, while in higher dimensions non-special solutions exist.

Contribution

It characterizes the behavior of radial solutions across dimensions, showing the uniqueness in two dimensions and the existence of non-special solutions in higher dimensions.

Findings

All radial solutions in 2D are special Lagrangian.

In higher dimensions, non-special Lagrangian solutions exist.

Continuity of the gradient graph near the origin characterizes special Lagrangian solutions.

Abstract

We consider smooth radial solutions to the Hamiltonian stationary equation which are defined away from the origin. We show that in dimension two all radial solutions on unbounded domains must be special Lagrangian. In contrast, for all higher dimensions there exist non-special Lagrangian radial solutions over unbounded domains; moreover, near the origin, the gradient graph of such a solution is continuous if and only if the graph is special Lagrangian.