Existence of smooth even solutions to the dual Orlicz-Minkowski problem

Chen, Li; Liu, Yan‐Nan; Lu, Jiwen; Ni, Xiang

doi:10.48550/arxiv.2005.02639

Cited by 4 publications

(6 citation statements)

References 29 publications

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“…Our approach is based on the study of suitably designed parabolic flows. The flow technique has been proved to be effective and powerful in solving the Minkowski type and Gauss image problems [7,8,13,14,15,20,37,39,40,53]. The idea behind the flow technique is the fact that the Minkowski type and Gauss image problems can be reformulated as a Monge-Ampère type equation on S n , and this indeed works for the (extended) Musielak-Orlicz-Gauss image problem due to equation (1.11) (see [31]).…”

Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 99%

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A flow approach to the Musielak-Orlicz-Gauss image problem

Li,

Sheng,

et al. 2021

Preprint

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In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure C G,Ψ,λ (Ω, •) of convex body Ω in R n+1 containing the origin (but the origin is not necessary in its interior). In particular, we provide solutions to the extended Musielak-Orlicz-Gauss image problem based on the study of suitably designed parabolic flows, and by the use of approximation technique (for general measures). Our parabolic flows involve two Musielak-Orlicz functions and hence contain many well-studied curvature flows related to Minkowski type problems as special cases. Our results not only generalize many previously known solutions to the Minkowski type and Gauss image problems, but also provide solutions to those problems in many unsolved cases.

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Section: Introduction and Overview Of The Main Resultsmentioning

confidence: 99%

“…As (1.12) involves the Musielak-Orlicz functions, such a flow could be named a Musielak-Orlicz-Gauss curvature flow, which is arguably the most general curvature flow related to Minkowski type and Gauss image problems and contains all previous well-studied flows [7,8,13,14,15,20,37,39,40,53] as its special cases.…”

Section: Define α *mentioning

confidence: 99%

A flow approach to the Musielak-Orlicz-Gauss image problem

Li,

Sheng,

et al. 2021

Preprint

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“…In this section, we continue to establish uniform upper and lower bounds for principal curvatures. These estimates can be obtained by considering proper auxiliary functions, see [10,13,36,37] for similar techniques. First, we need the following lemma.…”

Section: Uniform Bounds For Principal Curvaturesmentioning

confidence: 99%

Smooth solutions to the Gauss image problem

Chen¹,

Wu²,

Ni³

2020

Preprint

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“…Lastly, the general dual Orlicz-Minkowski problem also extends the dual Orlicz-Minkowski problems [65,69] and the Minkowski problem for Gaussian measures [32]. Solutions to the (general) dual Orlicz-Minkowski problem by using the techniques from partial differential equations can be found in [12,13,44].…”

Section: Introductionmentioning

confidence: 98%

On the Musielak-Orlicz-Gauss image problem

Huang,

Xing,

et al. 2021

Preprint

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In the present paper we initiate the study of the Musielak-Orlicz-Brunn-Minkowski theory for convex bodies. In particular, we develop the Musielak-Orlicz-Gauss image problem aiming to characterize the Musielak-Orlicz-Gauss image measure of convex bodies. For a convex body K, its Musielak-Orlicz-Gauss image measure, denoted by C Θ (K, •), involves a triple Θ = (G, Ψ, λ) where G and Ψ are two Musielak-Orlicz functions defined on S n−1 × (0, ∞) and λ is a nonzero finite Lebesgue measure on the unit sphere S n−1 . Such a measure can be produced by a variational formula of V G,λ (K) (the general dual volume of K with respect to λ) under the perturbations of K by the Musielak-Orlicz addition defined via the function Ψ. The Musielak-Orlicz-Gauss image problem contains many intensively studied Minkowski type problems and the recent Gauss image problem as its special cases. Under the condition that G is decreasing on its second variable, the existence of solutions to this problem is established.

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Existence of smooth even solutions to the dual Orlicz-Minkowski problem

Cited by 4 publications

References 29 publications

A flow approach to the Musielak-Orlicz-Gauss image problem

A flow approach to the Musielak-Orlicz-Gauss image problem

Smooth solutions to the Gauss image problem

On the Musielak-Orlicz-Gauss image problem

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