A minimum principle for Lagrangian graphs

TL;DR

This paper introduces a new minimum principle in Lagrangian geometry connecting classical and space-time Lagrangian angles, leading to novel formulas for solutions of degenerate special Lagrangian equations.

Contribution

It establishes a minimum principle in Lagrangian geometry linking different Lagrangian angles and provides a new formula for degenerate special Lagrangian equations using partial Legendre transforms.

Findings

Established a minimum principle relating classical and space-time Lagrangian angles.

Derived a new formula for solutions of degenerate special Lagrangian equations.

Connected Lagrangian geometry with obstacle problems in PDEs.

Abstract

The classical minimum principle is foundational in convex and complex analysis and plays an important role in the study of the real and complex Monge-Ampere equations. This note establishes a minimum principle in Lagrangian geometry. This principle relates the classical Lagrangian angle of Harvey-Lawson and the space-time Lagrangian angle introduced recently by Rubinstein-Solomon. As an application, this gives a new formula for solutions of the degenerate special Lagrangian equation in space-time in terms of the (time) partial Legendre transform of a family of solutions of obstacle problems for the (space) non-degenerate special Lagrangian equation.