A boundary value problem for minimal Lagrangian graphs

TL;DR

This paper proves the existence of a diffeomorphism between two convex domains in Euclidean space such that its graph forms a minimal Lagrangian submanifold, solving a specific boundary value problem.

Contribution

It establishes the existence of minimal Lagrangian graphs with prescribed boundary conditions between convex domains, advancing geometric analysis.

Findings

Existence of a diffeomorphism with a minimal Lagrangian graph

Solution to a boundary value problem for minimal Lagrangian graphs

Extension of geometric PDE techniques to convex domains

Abstract

Let \Omega and \tilde{\Omega} be uniformly convex domains in \mathbb{R}^n with smooth boundary. We show that there exists a diffeomorphism f: \Omega \to \tilde{\Omega} such that the graph \Sigma = \{(x,f(x)): x \in \Omega\} is a minimal Lagrangian submanifold of \mathbb{R}^n \times \mathbb{R}^n.