On the second boundary value problem for Lagrangian mean curvature type equation

TL;DR

This paper studies a fully nonlinear parabolic equation related to special Lagrangian geometry, proving long-term existence, convergence, and uniqueness of solutions for a boundary value problem.

Contribution

It generalizes Brendle--Warren's theorem by establishing existence and uniqueness of smooth solutions for a Lagrangian mean curvature type equation with boundary conditions.

Findings

Proved long-time existence and convergence of the flow.

Established existence and uniqueness of smooth solutions.

Generalized previous results on minimal Lagrangian diffeomorphisms.

Abstract

This article is concerned with the second boundary value problem of the Lagrangian mean curvature type equation arising from special Lagrangian geometry. By the parabolic method, we consider a fully nonlinear parabolic equation with oblique derivative boundary condition, and show the long time existence and convergence of the flow. It follows that the existence and uniqueness of the smooth uniformly convex solution are obtained, which generalizes the Brendle--Warren's theorem about minimal Lagrangian diffeomorphism in Euclidean metric space.