Doubling and the two-dimensional critical valued Lagrangian phase
Analysis of PDEs
2025-10-28 v1 Differential Geometry
Abstract
In this paper, we establish interior Hessian and gradient estimates for the two-dimensional Lagrangian mean curvature equation when the phase changes signs, provided the gradient of the phase vanishes along its zero set. At the critical phase in two dimensions, the Jacobi inequality degenerates, preventing the use of higher-dimensional methods to obtain Hessian estimates. To address this difficulty, we introduce a modified doubling technique that applies to degenerate Jacobi inequalities and yields interior estimates.
Cite
@article{arxiv.2510.22887,
title = {Doubling and the two-dimensional critical valued Lagrangian phase},
author = {Arunima Bhattacharya and Ravi Shankar and Jeremy Wall},
journal= {arXiv preprint arXiv:2510.22887},
year = {2025}
}