Failure of Interior H\"older and Gradient Estimates for the Subcritical Special Lagrangian Equation

Abstract

For every $n\ge3$, we show, by a family of explicit entire real-analytic solutions, that, for every subcritical phase and every $0<\alpha\le1$, oscillation-based interior $C^\alpha$ estimates fail for the special Lagrangian equation in $\mathbb{R}^n$. In fact, this family has no common interior modulus of continuity.