The special Lagrangian equation is a fully nonlinear, elliptic PDE discovered by Harvey and Lawson in the 1980’s.  They showed that if solutions are C^{1,1}, then they correspond to minimal surfaces.  Recently, Mooney and Savin constructed Lipschitz viscosity solutions which are not minimal.

When can one rule out such non-minimal solutions?  The singular solutions constructed to date are semi-convex.  In joint work with Mooney, we find an optimal semi-convexity threshold for viscosity solutions.  Above this threshold, there is regularity, and below this threshold, non-minimal solutions show up.  The threshold preserves rough solutions under geometric changes of variables.  It is also connected to area-decreasing maps, despite those not having a viscosity formulation.