Counterexamples to the comparison principle in the special Lagrangian potential equation

TL;DR

This paper constructs counterexamples for the special Lagrangian potential equation, demonstrating that the comparison principle does not hold universally for continuous phases, which was previously an open question.

Contribution

It provides explicit counterexamples showing the failure of the comparison principle in the special Lagrangian potential equation for certain continuous phases.

Findings

Counterexamples show comparison principle fails for some continuous phases.

The constructed solutions have isolated maxima at the origin.

Addresses an open question about the validity of the comparison principle.

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.