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Advanced Nonlinear Studies

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Volume 18, Issue 2

Issues

Refined Boundary Behavior of the Unique Convex Solution to a Singular Dirichlet Problem for the Monge–Ampère Equation

Zhijun Zhang
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  • School of Mathematics and Information Science, Yantai University, Yantai 264005, Shandong, P. R. China
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Published Online: 2018-01-10 | DOI: https://doi.org/10.1515/ans-2017-6045

Abstract

This paper is concerned with the boundary behavior of the unique convex solution to a singular Dirichlet problem for the Monge–Ampère equation

detD2u=b(x)g(-u),u<0,xΩ,u|Ω=0,

where Ω is a strictly convex and bounded smooth domain in N, with N2, gC1((0,),(0,)) is decreasing in (0,) and satisfies lims0+g(s)=, and bC(Ω) is positive in Ω, but may vanish or blow up on the boundary. We find a new structure condition on g which plays a crucial role in the boundary behavior of such solution.

Keywords: Monge–Ampère Equations; Singular Boundary Value Problem; Convex Solutions, Boundary Behavior

MSC 2010: 35J25; 35J65; 35J67

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About the article


Received: 2017-01-09

Revised: 2017-12-06

Accepted: 2017-12-08

Published Online: 2018-01-10

Published in Print: 2018-04-01


Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11571295

This work is supported in part by NSF of P. R. China under grant 11571295.


Citation Information: Advanced Nonlinear Studies, Volume 18, Issue 2, Pages 289–302, ISSN (Online) 2169-0375, ISSN (Print) 1536-1365, DOI: https://doi.org/10.1515/ans-2017-6045.

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