On the $$L^{p}$$ dual Minkowski problem for $$-1<0$$

IF 2.1 2区数学 Q1 MATHEMATICS

Calculus of Variations and Partial Differential Equations Pub Date : 2024-09-17 DOI:10.1007/s00526-024-02806-5

Stephanie Mui

引用次数: 0

Abstract

The $L^{p}$ dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the $L^{p}$ dual Minkowski problem, asks about existence of a convex body with prescribed $L^{p}$ dual curvature measure. This question unifies the previously disjoint $L^{p}$ Minkowski problem with the dual Minkowski problem, two open questions in convex geometry. In this paper, we prove the existence of a solution to the $L^{p}$ dual Minkowski problem for the case of $q<p+1$, $-1<p<0$, and $p\ne q$ for even measures.

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关于$$-1<0$$的$$L^{p}$$对偶闵科夫斯基问题

Lutwak 等人提出了 $L^{p}$ 对偶曲率量（Adv Math 329:85-132, 2018）。相关的闵科夫斯基问题被称为$L^{p}$ 对偶闵科夫斯基问题，询问是否存在具有规定的$L^{p}$ 对偶曲率度量的凸体。这个问题统一了之前不相交的 $L^{p}$ Minkowski 问题和对偶 Minkowski 问题，这是凸几何中的两个悬而未决的问题。在本文中，我们证明了在$q<p+1$、$-1<p<0$和$p\ne q$ 偶数度量的情况下，$L^{p}$对偶 Minkowski 问题解的存在。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Calculus of Variations and Partial Differential Equations 数学-数学

CiteScore

3.30

自引率

4.80%

发文量

224

审稿时长

6 months

期刊介绍： Calculus of variations and partial differential equations are classical, very active, closely related areas of mathematics, with important ramifications in differential geometry and mathematical physics. In the last four decades this subject has enjoyed a flourishing development worldwide, which is still continuing and extending to broader perspectives. This journal will attract and collect many of the important top-quality contributions to this field of research, and stress the interactions between analysts, geometers, and physicists. The field of Calculus of Variations and Partial Differential Equations is extensive; nonetheless, the journal will be open to all interesting new developments. Topics to be covered include: - Minimization problems for variational integrals, existence and regularity theory for minimizers and critical points, geometric measure theory - Variational methods for partial differential equations, optimal mass transportation, linear and nonlinear eigenvalue problems - Variational problems in differential and complex geometry - Variational methods in global analysis and topology - Dynamical systems, symplectic geometry, periodic solutions of Hamiltonian systems - Variational methods in mathematical physics, nonlinear elasticity, asymptotic variational problems, homogenization, capillarity phenomena, free boundary problems and phase transitions - Monge-Ampère equations and other fully nonlinear partial differential equations related to problems in differential geometry, complex geometry, and physics.