The two dimensional \(L_p\) Minkowski problem and nonlinear equations with negative exponents
type: Item
instance of: scholarly article
title: The two dimensional \(L_p\) Minkowski problem and nonlinear equations with negative exponents
zbMATH Open document ID: 1260.52005
author: Jingbo Dou
author: Meijun Zhu
published in: Advances in Mathematics
publication date: 2012-07-20T00:00:00Z
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 6059027
zbMATH Keywords: \(L_{p}\) Minkowski problem
zbMATH Keywords: negative exponent
zbMATH Keywords: Blaschke-Santaló inequality
reviewed by: Maria A. Hernández Cifre
MaRDI profile type: Publication
full work available at URL: j.aim.2012.02.027
OpenAlex ID: W1965571776
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DOI: 10.1016/J.AIM.2012.02.027
review text: The \(L_p\) Minkowski problem asks when a given Borel measure on the sphere \(S^{n-1}\) can be the \(L_p\) surface area measure of a convex body \(K\). In the plane, this question is equivalent to the following analytical one: given a function \(g(\theta)\), are there positive solutions of the differential equation NEWLINE\[NEWLINE u''(\theta)+u(\theta)=g(\theta)u(\theta)^{p-1}\;\text{ on }\;S^1? NEWLINE\]NEWLINE In [Adv. Math. 201, No. 1, 77--89 (2006; Zbl 1102.34023)], \textit{W. Chen} obtained certain existence results for this problem when \(-2\leq p\leq 0\). In the paper under review the authors extend these results for all \(p\leq -2\). They also consider the related conformal curvature problem with opposite curvature sign, i.e., they give conditions for the existence of positive solutions of the differential equation \(-u''(\theta)+u(\theta)=g(\theta)u(\theta)^{p-1}\), \(p\leq 0\), on the 1-dimensional sphere.
description: scientific article; zbMATH DE number 6059027