Regularity for Hamiltonian stationary equations in ${\mathbb{R}}_{n\le 4}^n$

Abstract

In this paper, we study the regularity of solutions to the Hamiltonian stationary equation in complex Euclidean space. We show that in dimensions $n\le 4$, for all values of the Lagrangian phase, any $C^{1,1}$ solution is smooth and derive a $C^{k,\alpha }$ estimate for it, where $k\ge 2$.