A Boundary Value Problem for Minimal Lagrangian Graphs Simon Brendle and Micah Warren

A submanifold Σ ⊂ Rn × Rn is called Lagrangian if ω|Σ = 0. In this paper, we study a boundary value problem for minimal Lagrangian graphs in Rn×Rn. To that end, we fix two domains Ω, Ω̃ ⊂ Rn with smooth boundary. For each diffeomorphism f : Ω → Ω̃, we consider its graph Σ = {(x, f(x)) : x ∈ Ω} ⊂ Rn × Rn. We consider the problem of finding a diffeomorphism f : Ω → Ω̃ such that Σ is Lagrangian and has zero mean curvature. Our main result asserts that this is possible if Ω and Ω̃ are strictly convex: Theorem. Let Ω and Ω̃ be strictly convex domains in Rn with smooth boundary. Then there exists a diffeomorphism f : Ω → Ω̃ such that the graph