Diff. Geom, Math. Phys., PDE Seminar: Ravi Shankar

Regularity has long been known to fail for minimal submanifolds of Euclidean space with large dimension, or large codimension, the latter in part because of no maximum principle for systems of PDEs. If, however, a minimal submanifold of Euclidean 2n-space has the gradient form (x,Du(x)) for a (Lagrangian) potential u(x), then u(x) satisfies a single elliptic PDE with a maximum principle, called the special Lagrangian equation (SLE), so there is hope for a regularity theory of continuous (viscosity) solutions. At the same time, the equation itself says the phase function, or the trace of arctan D^2u(x), is constant, so the PDE fails the uniform ellipticity condition exemplified by the ordinary trace of D^2u(x), which precludes standard methods in regularity theory. As a compromise, for large phases, solutions are smooth in the interior of a domain, and for smaller phases, counterexamples exist. One last puzzle remained: in 2009, [Chen-Warren-Yuan] showed a regularity estimate assuming instead smoothness and convexity. The convexity condition is necessary given semiconvex counterexamples, but the smoothness assumption could not be weakened to merely convex viscosity solutions. In our preprint [Chen-Shankar-Yuan], we establish interior regularity for convex viscosity solutions of the SLE. Our approach combines elementary convex duality, geometric measure theory, and maximum principle ideas