Notes on generalized special Lagrangian equation
type: Item
instance of: scholarly article
title: Notes on generalized special Lagrangian equation
author: Xingchen Zhou
published in: Calculus of Variations and Partial Differential Equations
publication date: 2024-08-15T00:00:00Z
review text: The author is interested to the generalized Lagrangian phase equation \N\[\N\sum_{i=1}^n\arctan\frac{\lambda_i(D^2u)}{f(x)}=\Theta,\N\]\Nin dimension \(n \ge 3\), where \(\lambda_i\) are the eigenvalues of the Hessian matrix \(D^2u\), \(f\) is a positive Lipschitz continuous function and \(\Theta\) the phase constant.\N\NIn particular, the range \(|\Theta| \ge \frac\pi2\,(n-2)\) is considered. In this case, local gradient and Hessian bounds are derived for positive and smooth functions \(f\).\N\NExistence and uniqueness of \(C^{2,\alpha}\) solutions is shown for Dirichlet problems with continuous boundary data when \(f\) is Lipschitz, \(f \ge 1\), and \(\Theta \in [\frac\pi2\,(n-2),\frac\pi2\,n)\), based on an argument due to Caffarelli-Nirenberg-Spruck and Yuan, by using Schauder theory and \(C^{2,\alpha}\) estimates for smooth \(f\).\N\NAs an application, a \(C^{2,\alpha}\) regularity result for continuous viscosity solutions of equation \(\sigma_2(D^2u)=f^2(x)\) with a positive and Lipschitz continuous \(f\) is stated in dimension \(n=3\).
reviewed by: Antonio Vitolo
Mathematics Subject Classification ID: 35B45
Mathematics Subject Classification ID: 35B65
Mathematics Subject Classification ID: 35D40
zbMATH DE Number: 7897401
zbMATH Keywords: special Lagrangian equations
zbMATH Keywords: positive Lipschitz right hand sides
zbMATH Keywords: Hessian quotient equations
MaRDI profile type: Publication
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DOI: 10.1007/S00526-024-02801-W
description: scientific article; zbMATH DE number 7897401