The Neumann problem of special Lagrangian type equations

TL;DR

This paper investigates the Neumann boundary value problem for special Lagrangian type equations with critical and supercritical phases, establishing existence through new boundary estimates and gradient bounds.

Contribution

It provides a direct proof of boundary double normal derivative estimates and solves the classical Neumann problem for these equations.

Findings

Established uniform second order a priori estimates

Proved boundary double normal derivative estimates directly

Solved the classical Neumann problem with gradient estimates

Abstract

We study the Neumann problem for special Lagrangian type equations with critical and supercritical phases. These equations naturally generalize the special Lagrangian equation and the k-Hessian equation. By establishing uniform a priori estimates up to the second order, we obtain the existence result using the continuity method. The new technical aspect is our direct proof of boundary double normal derivative estimates. In particular, we directly prove the double normal estimates for the 2-Hessian equation in dimension 3. Moreover, we solve the classical Neumann problem by proving the uniform gradient estimate.