Counterexamples to the comparison principle in the special Lagrangian potential equation

For each k = 0 , ⋯ , n we construct a continuous phase f k , with f k ( 0 ) = ( n - 2 k ) π 2 , and viscosity sub- and supersolutions v k , u k , of the elliptic PDE ∑ i = 1 n arctan ( λ i ( 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 : R n ⊇ Ω → ( - n π / 2 , n π / 2 ) . Our examples show it does not.