$\mathbb{R}$2上对偶Minkowski问题的可解性

Abstract

Abstract:

In this paper, we study the existence of minimum of a constrained variational problem in the Sobolev space W^{1, 4}($\mathbb{S}$). If ∫$_\mathbb{S}$g(θ)dθ>0, the minimum is a positive solution to the related Euler-Lagrange equation

${u^{\prime \prime }} + u = \frac{{g(\theta )}}{{u\left( {{u^2} + {u^{\prime 2}}} \right)}}, \theta \in {\rm{\mathbb{S}}}$

Based on this, we prove the solvability of the dual Minkowski problem in $\mathbb{R}$² posed by Huang-Lutwak-Yang-Zhang[Acta Math, 2016, 216(2):325-338].

Key words: Dual Minkowski problem, Nonlinear equation, Variational method

CLC Number:

O176

Na Wei. The Solvability of Dual Minkowski Problem in $\mathbb{R}$²[J].Acta mathematica scientia,Series A, 2019, 39(6): 1314-1322.

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References 24

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The Solvability of Dual Minkowski Problem in $\mathbb{R}$²

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The Solvability of Dual Minkowski Problem in $\mathbb{R}$²