Optimal Regularity for Lagrangian Mean Curvature Type Equations

We classify regularity for Lagrangian mean curvature type equations, which include the potential equation for prescribed Lagrangian mean curvature and those for Lagrangian mean curvature flow self-shrinkers and expanders, translating solitons, and rotating solitons. Convex solutions of the second boundary value problem for certain such equations were constructed by Brendle-Warren (J Differ Geom 84(2):267-287, 2010), Huang (J Funct Anal 269(4):1095-1114, 2015), and Wang-Huang-Bao (Calc Var Partial Differ Equ 62(3):74 2023). We first show that convex viscosity solutions are regular provided the Lagrangian angle or phase is \(C^2\) $C^2$ and convex in the gradient variable. We next show that for merely Hölder continuous phases, convex solutions are regular if they are \(C^{1,\beta }\) $C^{1,\beta }$ for sufficiently large \(\beta \) $\beta$ . Singular solutions are given to show that each condition is optimal and that the Hölder exponent is sharp. Along the way, we generalize the constant rank theorem of Bian and Guan to include arbitrary dependence on the Legendre transform.