#### Vol. 318, No. 1, 2022

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A note on the two-dimensional Lagrangian mean curvature equation

### Arunima Bhattacharya

Vol. 318 (2022), No. 1, 43–50
DOI: 10.2140/pjm.2022.318.43
##### Abstract

We use Warren–Yuan’s (Warren and Yuan 2009) superisoperimetric inequality on the level sets of subharmonic functions, which is available only in two dimensions, to derive a modified Hessian bound for solutions of the two-dimensional Lagrangian mean curvature equation. We assume the Lagrangian phase to be supercritical with bounded second derivatives. Unlike in (Bhattacharya 2021), the simplified approach in this paper does not require the Michael–Simon mean value and Sobolev inequalities on generalized submanifolds of ${ℝ}^{n}$ (Michael and Simon 1973).

##### Keywords
Lagrangian mean curvature, critical phase, supercritical phase, Hessian estimates
Primary: 35B45
Secondary: 35J60
##### Milestones
Received: 19 October 2021
Revised: 14 April 2022
Accepted: 16 April 2022
Published: 1 August 2022
##### Authors
 Arunima Bhattacharya Department of Mathematics University of Washington Seattle, WA United States