Concavity of the Lagrangian Phase Operator and Applications

2016/07/25 by Tristan C. Collins, Sébastien Picard, Collins, Tristan C. +3 · 5 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.1607.07194

openalex publication_date 2016/07/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if Ω is a compact domain in ℝⁿ or ℂⁿ, then there exists a solution to the Dirichlet problem with right-hand side h(x) satisfying |h(x)| > (n-2)\fracπ2 and boundary data ϕ if and only if there exists a subsolution.

Citations

• (1,1) forms with specified Lagrangian phase: A priori estimates and\n algebraic obstructions
• Doubly Stochastic Matrices and the Diagonal of a Rotation Matrix

Cited by

• On the smoothness of solutions of fully nonlinear second order equations in the plane
• Doubling and the two-dimensional critical valued Lagrangian phase
• Optimal Regularity for Hölder continuous Hamiltonian Stationary Lagrangian graphs
• Singularities of the Lagrangian mean curvature flow at the critical Lagrangian phase
• Semi-convex viscosity solutions of the special Lagrangian equation

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• (1,1) forms with specified Lagrangian phase: A priori estimates and\n algebraic obstructions
• A double phase problem involving Hardy potentials
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