Rotationally symmetric solutions to the \(L_p\)-Minkowski problem
type: Item
instance of: scholarly article
title: Rotationally symmetric solutions to the \(L_p\)-Minkowski problem
zbMATH Open document ID: 1273.52006
author: Jian Lu
author: Xu-Jia Wang
published in: Journal of Differential Equations
publication date: 2013-01-18T00:00:00Z
review text: Let \(f\) be a smooth positive function on the unit sphere \(\mathbb{S}^n\) in \(\mathbb{R}^{n+1}\). The authors study the equation \[ \det (\nabla^2 H +HI) = \frac{f}{H^{n+2}} \quad\text{on}\quad \mathbb{S}^n \leqno(1) \] where \(H\) is the support function of a bounded convex body \(K\) in \(\mathbb{R}^{n+1}\), \(I\) is the identity matrix, and \(\nabla^2 H\) denotes the covariant second derivatives of \(H\) with respect to an orthonormal frame on \(\mathbb{S}^n\). This equation arises in several problems, including the \(L_p\)-Minkowski problem with \(p=n-1\) studied by \textit{E.~Lutwak} [J. Differ. Geom. 38, No. 1, 131--150 (1993; Zbl 0788.52007)], the centroaffine Minkowski problem studied by \textit{K.-S.~Chou} and \textit{X.-J.~Wang} [Adv. Math. 205, No. 1, 33--83 (2006; Zbl 1245.52001)], self-similar solutions of anisotropic curve shortening flow studied by \textit{B.~Andrews} [Calc. Var. Partial Differ. Equ. 7, No. 4, 315--371 (1998; Zbl 0931.53030)] and \textit{K.-S.~Chou} and \textit{X.-P.~Zhu} [The curve shortening problem. Boca Raton, FL: Chapman \& Hall/CRC (2001; Zbl 1061.53045)] in the case \(n=1\), and in connection with the critical case of the Blaschke-Santalo inequality studied by \textit{E.~Lutwak} and \textit{G.~Zhang} [J. Differ. Geom. 47, No. 1, 1--16 (1997; Zbl 0906.52003)]. The existence problem for this equation is rather delicate. First, when \(f\) is constant, the support function of any ellipsoid centred at the origin is a solution of \((1)\). So one cannot obtain a priori estimates for solutions without additional assumptions on \(f\). Second, there exists a Kazdan-Warner type obstruction to the existence of solutions, namely for any \((n+1)\times(n+1)\) matrix \(B\), the vector field \(\xi(x)=Bx-(x^TBx)x\), \(x\in\mathbb{S}^n\), must satisfy \[ \int_{\mathbb{S}^n} \frac{\nabla_\xi f}{H^{n+1}}=0 \leqno(2) \] whenever \(H\) is a \(C^3\) solution of \((1)\). The authors prove the following results. Theorem 1. There exist positive consta…
Mathematics Subject Classification ID: 52A20
Mathematics Subject Classification ID: 35J96
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 6126877
zbMATH Keywords: Monge-Ampère equation
zbMATH Keywords: a priori estimates
zbMATH Keywords: existence of solutions
zbMATH Keywords: \(L_p\)-Minkowski problem
zbMATH Keywords: rotationally symmetric solutions
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: j.jde.2012.10.008
OpenAlex ID: W2072559369
DOI: 10.1016/J.JDE.2012.10.008
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description: scientific article; zbMATH DE number 6126877