{"title":"On the $$L^{p}$$ dual Minkowski problem for $$-1<0$$","authors":"Stephanie Mui","doi":"10.1007/s00526-024-02806-5","DOIUrl":null,"url":null,"abstract":"<p>The <span>\\(L^{p}\\)</span> dual curvature measure was introduced by Lutwak et al. (Adv Math 329:85–132, 2018). The associated Minkowski problem, known as the <span>\\(L^{p}\\)</span> dual Minkowski problem, asks about existence of a convex body with prescribed <span>\\(L^{p}\\)</span> dual curvature measure. This question unifies the previously disjoint <span>\\(L^{p}\\)</span> 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 <span>\\(L^{p}\\)</span> dual Minkowski problem for the case of <span>\\(q<p+1\\)</span>, <span>\\(-1<p<0\\)</span>, and <span>\\(p\\ne q\\)</span> for even measures.\n</p>","PeriodicalId":9478,"journal":{"name":"Calculus of Variations and Partial Differential Equations","volume":"11 14 1","pages":""},"PeriodicalIF":2.1000,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Calculus of Variations and Partial Differential Equations","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00526-024-02806-5","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
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.
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.