On the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase

TL;DR

This paper addresses the Dirichlet problem for the Lagrangian phase equation in critical and supercritical phases, establishing interior estimates and demonstrating the existence of singular solutions in subcritical cases.

Contribution

It provides the first sharp interior $C^2$ estimates for the problem and highlights the existence of singular solutions in subcritical phases.

Findings

Established sharp interior $C^2$ estimates.

Proved existence of singular solutions in subcritical phase.

Solved the Dirichlet problem for critical and supercritical phases.

Abstract

In this paper, we solve the Dirichlet problem for Lagrangian phase equation with critical and supercritical phase. A crucial ingredient is the interior $C^2$ estimate. Our result is sharp in the sense that there exist singular solutions in the subcritical phase case.