Sharp Boundary Recession Criteria for the Special Lagrangian Curvature Potential Equation

Abstract

We establish boundary second derivative estimates for convex graphical solutions of the special Lagrangian curvature potential equation. Since the curvature matrix depends on both $Du$ and $D^{2}u$, a phase subsolution alone does not provide the full linearized separation needed for the mixed derivative estimate. We introduce a mixed recession compatibility condition imposed only on doubly degenerate level jets. It yields a uniform mixed derivative bound and is sharp within the class of fixed smooth zero-order barriers considered here. For the double-normal derivative, an exact complex Schur-complement identity gives \[ u_{νν}=β+α\cotδ, \qquad 1\leqα\leq C, \qquad |β|\leq C, \] where $\alpha$ and $\beta$ are explicit Schur-complement coefficients, $\delta$ is the actual boundary limiting-phase gap, and $C$ depends only on uniform bounds for the gradient and the mixed boundary derivatives. Thus curvature blows up if and only if this gap collapses, with optimal rate ${\delta }^{-1}$. Smooth radial solutions attain the rate, while a rank-loss model shows that a strict lower subsolution need not force strict convexity.

32 pages, 2 figures