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 $f\colon\Omega\to (-n\pi/2,n\pi/2)$. Our examples show it does not.