Counterexamples to the comparison principle in the special Lagrangian potential equation

Abstract

For each $k=0,\dots ,n$ we construct a continuous phase $f_k$, with $f_k(0)=(n-2k)\frac{\pi }{2}$, and viscosity sub- and supersolutions $v_k$, $u_k$, of the elliptic PDE ${\sum }_{i=1}^n\arctan ({\lambda }_i(D^{2}w))=f_k(x)$ such that $v_k-u_k$ has an isolated maximum at the origin. It has been an open question whether the comparison principle would hold in this second order equation for arbitrary continuous phases $f:\Omega \to (-n\pi /2,n\pi /2)$. Our examples show it does not.