Counterexamples to the comparison principle in the special Lagrangian potential equation

For each \(k = 0,\dots ,n\) $k=0,\cdots ,n$ we construct a continuous phase \(f_k\) $f_k$ , with \(f_k(0) = (n-2k)\frac{\pi }{2}\) $f_k(0)=(n-2k)\frac{\pi }{2}$ , and viscosity sub- and supersolutions \(v_k\) $v_k$ , \(u_k\) $u_k$ , of the elliptic PDE \(\sum _{i=1}^n \arctan (\lambda _i(\mathcal {H}w)) = f_k(x)\) ${\sum }_{i=1}^{n}arctan({\lambda }_i(\mathcal{H}w))=f_k(x)$ such that \(v_k-u_k\) $v_k-u_k$ 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 \(f:\mathbb {R}^n\supseteq \Omega \rightarrow (-n\pi /2,n\pi /2)\) $f:{\mathbb{R}}^n\supseteq \mathrm{\Omega }\to (-n\pi /2,n\pi /2)$ . Our examples show it does not.