The obstacle problem for the Monge-Ampère equation with the lower obstacle
type: Item
instance of: scholarly article
title: The obstacle problem for the Monge-Ampère equation with the lower obstacle
zbMATH Open document ID: 1475.35169
author: Ki-Ahm Lee
author: Taehun Lee
author: Jinwan Park
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2021-06-14T00:00:00Z
review text: The authors consider the existence and optimal regularity of the solution and the regularity of the free boundary of the obstacle problem for the following equation involving the log-concave Monge-Ampère operator \[ \begin{cases}{c}\max \left\{ \varphi -u,\det D^{2}u-f(x,u,Du)\right\} =0\quad&\text{in }\Omega,\\ u=g&\text{in }\partial \Omega, \end{cases} \] with a strictly convex lower obstacle function \(\varphi \in C^{2,1}\left( \overline{\Omega }\right) \). Here \(\Omega \) is a strictly convex bounded domain in \(\mathbb{R}^{n}\) with \(\partial \Omega \in C^{3,1}\). About the forcing term \(f\) it is assumed that \(f\in C^{2,1}\left( \overline{\Omega }\times\mathbb{R}\times\mathbb{R}^{n}\right) \), \(f(\overline{\Omega }\times \lbrack -M,M]\times \lbrack-M,M]^{n})\subset [\sigma _{1}(M),\sigma _{2}(M)]\) (\(-\infty <\sigma _{1}\leq \sigma _{2}<+\infty \)), and the the boundary data \(g\in C^{3,1}\left( \partial \Omega \right) \) is such that \(\varphi <g\) on \(\partial\Omega\). The obstacle problem for the Monge-Ampère equation arises in the study of the prescribed Gaussian curvature equation, optimal transportation, and affine geometry with an obstacle. The incongruousness of the operator and the location of the obstacle below the solution cause various difficulties in the penalization method, the classification of the global solutions, and the directional monotonicity.
reviewed by: Marek Galewski
Mathematics Subject Classification ID: 35J96
Mathematics Subject Classification ID: 35A01
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 7358557
zbMATH Keywords: obstacle problem
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: existence
zbMATH Keywords: regularity
MaRDI profile type: Publication
full work available at URL: j.na.2021.112374
OpenAlex ID: W3162242661
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DOI: 10.1016/J.NA.2021.112374
description: scientific article; zbMATH DE number 7358557