Counterexamples to the comparison principle in the special Lagrangian potential equation

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 \(\sum _{i=1}^n \arctan (\lambda _i(\mathcal {H}w)) = f_k(x)\) 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 \(f:\mathbb {R}^n\supseteq \Omega \rightarrow (-n\pi /2,n\pi /2)\). Our examples show it does not.