YanNan Liu, Jian Lu
In this paper the dual Orlicz-Minkowski problem, a generalization of the $L_p$ dual Minkowski problem, is studied. By studying a flow involving the Gauss curvature and support function, we obtain a new existence result of solutions to this problem for smooth measures.
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In this paper the Orlicz-Minkowski problem for torsional rigidity, a generalization of the classical Minkowski problem, is studied. Using the flow method, we obtain a new existence result of solutions to this problem for general measures.
In this paper we study the dual Orlicz-Minkowski problem, which is a generalization of the dual Minkowski problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain a new existence result of smooth even solutions to this problem for smooth even measures.
In this paper a generalized Gauss curvature flow about a convex hypersurface in the Euclidean $n$-space is studied. This flow is closely related to the Orlicz-Minkowski problem, which involves Gauss curvature and a function of support function. Under some appropriate assumptions, we prove the long-time existence and convergence of this flow. As a byproduct, two existence results of solutions to the even Orlicz-Minkowski problem are obtained, one of which improves the known result.
In this paper, we study the long-time existence and asymptotic behavior for a class of anisotropic inverse Gauss curvature flows. By the stationary solutions of anisotropic flows, we obtain some new existence results for the dual Orlicz Minkowski type problem and even dual Orlicz Minkowski type problem for smooth measures, which is the most reasonable extension of the $L^p$ dual Minkowski problem from the dual point of view. The results of corresponding $L^p$ versions are $L^p$ dual Minkowski problem for $p>q$; and even $L^p$ dual Minkowski problem for $p>-1$, or $q<1$, or some ranges of $p<0<…
We study the long-time existence and behavior for a class of anisotropic non-homogeneous Gauss curvature flows whose stationary solutions, if exist, solve the regular Orlicz-Minkowski problems. As an application, we obtain old and new results for the regular even Orlicz-Minkowski problems; the corresponding $L_p$ version is the even $L_p$-Minkowski problem for $p>-n-1$. Moreover, employing a parabolic approximation method, we give new proofs of some of the existence results for the general Orlicz-Minkowski problems; the $L_p$ versions are the even $L_p$-Minkowski problem for $p>0$ and the $L_p…
Flow by Gauss Curvature to the Orlicz Minkowski Problem for q-torsional rigidity
本文首次提出并研究 q-扭转刚度的 Orlicz Minkowski 问题，利用高斯曲率流方法证明了 q>1 时光滑非偶解的存在性，填补了该领域的研究空白。
Anisotropic flows without global terms and dual Orlicz Christoffel-Minkowski type problem
本文研究了无全局项的各向异性曲率流的长时间存在性，并利用其平稳解解决对偶Orlicz Christoffel-Minkowski型问题，涵盖多种已知问题并部分回答了一般指定曲率问题。
An Orlicz version of the $L_p$-Minkowski problem on $S^{n-1}$ is discussed corresponding to the case $-n<p<0$.
In this paper, combining the $p$-capacity for $p\in (1, n)$ with the Orlicz addition of convex domains, we develop the $p$-capacitary Orlicz-Brunn-Minkowski theory. In particular, the Orlicz $L_ϕ$ mixed $p$-capacity of two convex domains is introduced and its geometric interpretation is obtained by the $p$-capacitary Orlicz-Hadamard variational formula. The $p$-capacitary Orlicz-Brunn-Minkowski and Orlicz-Minkowski inequalities are established, and the equivalence of these two inequalities are discussed as well. The $p$-capacitary Orlicz-Minkowski problem is proposed and solved under some mild…
Flow by Gauss Curvature to the orlicz Chord Minkowski Problem
本文研究 orlicz 弦 Minkowski 问题，通过用连续函数 $\varphi$ 替换 $p$ 推广了 $L_p$ 弦 Minkowski 问题，并利用高斯曲率流方法证明了其光滑解的存在性。
The general volume of a star body, a notion that includes the usual volume, the $q$th dual volumes, and many previous types of dual mixed volumes, is introduced. A corresponding new general dual Orlicz curvature measure is defined that specializes to the $(p,q)$-dual curvature measures introduced recently by Lutwak, Yang, and Zhang. General variational formulas are established for the general volume of two types of Orlicz linear combinations. One of these is applied to the Minkowski problem for the new general dual Orlicz curvature measure, giving in particular a solution to the Minkowski prob…
In this paper we study a normalised anisotropic Gauss curvature flow of strictly convex, closed hypersurfaces in the Euclidean space R^n+1. We prove that the flow exists for all time and converges smoothly to the unique, strictly convex solution of a Monge-Amp`ere type equation. Our argument provides a parabolic proof in the smooth category for the existence of solutions to the Dual Orlicz-Minkowski problem introduced by Zhu, Xing and Ye.
In this paper, the extended Musielak-Orlicz-Gauss image problem is studied. Such a problem aims to characterize the Musielak-Orlicz-Gauss image measure $\widetilde{C}_{G,Ψ,λ}(Ω,\cdot)$ of convex body $Ω$ in $\mathbb{R}^{n+1}$ 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-stud…
The general dual volume $\dveV(K)$ and the general dual Orlicz curvature measure $\deV(K, \cdot)$ were recently introduced for functions $G: (0, \infty)\times \sphere\rightarrow (0, \infty)$ and convex bodies $K$ in $\R^n$ containing the origin in their interiors. We extend $\dveV(K)$ and $\deV(K, \cdot)$ to more general functions $G: [0, \infty)\times \sphere\rightarrow [0, \infty)$ and to compact convex sets $K$ containing the origin (but not necessarily in their interiors). Some basic properties of the general dual volume and of the dual Orlicz curvature measure, such as the continuous depe…
For $K\subseteq \mathbb{R}^n$ a convex body with the origin $o$ in its interior, and $ϕ:\mathbb{R}^n\setminus\{o\}\rightarrow(0, \infty)$ a continuous function, define the general dual ($L_ϕ)$ Orlicz quermassintegral of $K$ by $$\mathcal{V}_ϕ(K)=\int_{\mathbb{R}^n \setminus K} ϕ(x)\,dx.$$ Under certain conditions on $ϕ$, we prove a variational formula for the general dual ($L_ϕ)$ Orlicz quermassintegral, which motivates the definition of $\widetilde{C}_{ϕ,\mathcal{V}}(K, \cdot)$, the general dual ($L_ϕ)$ Orlicz curvature measure of $K$. We pose the following general dual Orlicz-Minkowski probl…
An inverse Gauss curvature flow and its application to p-capacitary Orlicz-Minkowski problem
本文首次通过一类逆高斯曲率流证明了p∈(1,n)时p-容Orlicz-Minkowski问题光滑解的存在性，并在特殊情形下证明了解的唯一性。
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 $\widetilde{C}_Θ(K, \cdot)$, involves a triple $Θ=(G, Ψ, λ)$ where $G$ and $Ψ$ are two Musielak-Orlicz functions defined on $S^{n-1}\times (0, \infty)$ and $λ$ is a nonzero finite Lebesgue measure on the unit sphere $S^{n-1}$. Such a measure can be produced by a variati…
We study a general nonlinear elliptic equation in the Orlicz setting with data not belonging to the dual of the energy space. We provide several Lorentz-type and Morrey-type estimates for the gradients of solutions under various conditions on the data.
Uniqueness of solutions to a class of non-homogeneous curvature problems
本文利用 Heintze-Karcher 不等式证明一类各向同性混合 Christoffel-Minkowski 及 Minkowski 型问题的光滑凸解只能是中心球面，并推出 Orlicz-Minkowski 与 $L_p$-Gaussian-Minkowski 问题的唯一性。
The dual $L_p$-Minkowski problem with $p<0<q$ is investigated in this paper. By proving a new existence result of solutions and constructing an example, we obtain the non-uniqueness of solutions to this problem.
Existence of solutions to the generalized dual Minkowski problem
本文利用变分方法研究广义对偶Minkowski问题，分别解决了q<0及0≤q≤1的偶情形，并给出1<q<n时偶问题解存在的充分条件。
In this paper, we study $C^{1, 1}$ regularity for solutions to the degenerate $L_p$ Dual Minkowski problem. Our proof is motivated by the idea of Guan and Li's work on $C^{1,1}$ estimates for solutions to the Aleksandrov problem.
The uniqueness and non-uniqueness of solutions to the even dual Minkowski problem
该研究针对ℝⁿ中偶对偶曲率测度的Minkowski问题，利用新建立的对数Brunn-Minkowski不等式，证明了不同参数范围内解的唯一性或非唯一性结论。
The dual Minkowski problem for positive indices
该研究推导了偶情形下近常数密度对偶曲率测度的稳定性，以此为基础证明了正指标偶对偶Minkowski问题在ℝ^{n+1}中解的存在性与唯一性，要求测度密度在C^α范数下接近1。
The $L_p$ dual Minkowski problem for capillary hypersurfaces
本文研究了$L_p$对偶Minkowski问题，通过转化为Monge-Ampère方程并证明在特定条件下存在唯一解，解决毛细超曲面的曲率测度问题。
Recently, the $L_p$ dual Minkowski problem for unbounded closed convex sets in a pointed closed convex cone was proposed and a weak solution to this problem was provided. In smooth setting, this problem is equivalent to solving the Dirichlet problem for a class of Monge-Ampère type equations. In this paper, we show the existence, regularity and uniqueness of solutions to this Monge-Ampère type equation in the case $p\geq 1$ by studying variational properties for a family of Monge-Ampère functionals. Moreover, the existence and optimal global Hölder regularity in the case $p<1$ and $q\geq n$ is…
Orlicz space relaxation of total variation denoising
本文提出一种基于Orlicz空间松弛的全变分去噪模型，通过对数密度校正减少阶梯效应，并证明其收敛到ROF模型。
Diameter estimate for planar $L_p$ dual Minkowski problem
本文研究平面$L_p$对偶Minkowski问题，在$0<p<1$、$q\ge2$及测度密度有界正的条件下建立解的一致直径估计，还证明了测度密度在$C^\alpha$中接近常数时$L_p$ Minkowski问题解的唯一性与正性。
Functions of bounded Musielak-Orlicz-type deformation and anisotropic Total Generalized Variation for image-denoising problems
本文提出了一种具有各向异性总广义变分的图像去噪方法，结合有界定义场空间和广义Orlicz增长，用于改进图像去噪的数学模型和算法
In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ is the radial function, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When $ψ=1$, we prove that the flow exists for all time and converges to infinity if $α+δ+β\le1, β>0$ and $α\le0$, while in case $α+δ+β>1,α,δ\le0$, the flow blo…
We establish pointwise estimates expressed in terms of a nonlinear potential of a generalized Wolff type for $A$-superharmonic functions with nonlinear operator $A:Ω\times\mathbb{R}^n\to\mathbb{R}^n$ having measurable dependence on the spacial variable and Orlicz growth with respect to the last variable. The result is sharp as the same potential controls bounds from above and from below. Applying it we provide a bunch of precise regularity results including continuity and Hölder continuity for solutions to problems involving measures that satisfies conditions expressed in the natural scales. F…
The $L_{p}$ dual Christoffel-Minkowski problem for $1<p<q\leq k+1$ with $1\leq k\leq n$
本文研究 $L_p$ Christoffel-Minkowski 型问题，通过梯度估计证明了在 $1<p<q\leq k+1$ 条件下存在偶光滑严格凸解。
Under various conditions on the data we analyse how appearence of lower order terms affects the gradient estimates on solutions to a general nonlinear elliptic equation of the form \[-{\rm div}\, a(x,Du)+b(x,u)=μ\] with data $μ$ not belonging to the dual of the natural energy space but to Lorentz/Morrey-type spaces. The growth of the leading part of the operator is governed by a function of Orlicz-type, whereas the lower-order term satisfies the sign condition and is minorized with some convex function, whose speed of growth modulates the regularization of the solutions.
The solvability of the $L_p$-dual mixed Euclidean-Gauss Minkowski problem
本文研究$L_p$-对偶混合欧几里得-高斯闵可夫斯基问题，将其转化为单位球面上Monge-Ampère型方程，在$p<1$、$q<0$且$q<p$时证明解的存在性，在$p\geq q$时证明唯一性，并给出$p<1$、$q<0$时存在性的完整解答。
On the number of solutions to the planar dual Minkowski problem
本文研究二维对偶Minkowski问题，通过带参数积分的理论分析与数值估计，确定了常数对偶曲率情形下解的个数及非唯一性，并推广至$ L_p$-Alexandrov问题。
The capillary $L_{p}$ dual Christoffel-Minkowski problem for $1<p<q\leq k+1$ with $1\leq k\leq n$
本文针对毛细管 $L_p$ 对偶 Christoffel-Minkowski 问题，通过梯度估计，在 $1<p<q\leq k+1$ 参数范围内证明了偶、光滑、严格凸解的存在性。
In this paper we study a contracting flow of closed, convex hypersurfaces in the Euclidean space $\mathbb R^{n+1}$ with speed $f r^α K$, where $K$ is the Gauss curvature, $r$ is the distance from the hypersurface to the origin, and $f$ is a positive and smooth function. If $α\ge n+1$, we prove that the flow exists for all time and converges smoothly after normalisation to a soliton, which is a sphere centred at the origin if $f \equiv 1$. Our argument provides a parabolic proof in the smooth category for the classical Aleksandrov problem, and resolves the dual q-Minkowski problem introduced by…
The capillary $L_p$ dual Minkowski problem for $p>q$ in higher dimensions
本文针对高维空间中 p>q 的毛细管 L_p 对偶 Minkowski 问题，通过构造辅助函数并转化为带 Robin 边界条件的 Monge-Ampère 方程，证明了存在唯一光滑解。
The dual Minkowski problem for even data asks what are the necessary and sufficient conditions on an even prescribed measure on the unit sphere for it to be the $q$-th dual curvature measure of an origin-symmetric convex body in $\mathbb{R}^n$. A full solution to this is given when $1 < q < n$. The necessary and sufficient condition is an explicit measure concentration condition. A variational approach is used, where the functional is the sum of a dual quermassintegral and an entropy integral. The proof requires two crucial estimates. The first is an estimate of the entropy integral proved usi…
On the uniqueness of solutions to the isotropic $L_{p}$ dual Minkowski problem
本文证明了在特定参数范围内，单位球面是各向同性 $L_p$ 对偶 Minkowski 问题的唯一光滑严格凸解。
The $p$-th dual Minkowski problem for the $k$-torsional rigidity corresponding to a $k$-Hessian equation
本文提出第p次对偶Minkowski问题，将其转化为非线性偏微分方程，并证明了光滑解的存在性。
The $L_p$ dual Minkowski problem for $p>1$ and $q>0$
本文研究了$p>1$和$q>0$下的$L_p$对偶Minkowski问题的存在性，并讨论了解的正则性。
In his beautiful paper [1], Ben Andrews obtained the complete classification of the solutions of the planar isotropic $L_p$ Minkowski problem. In this paper, by generalizing Ben Andrews's result we obtain the complete classification of the solutions of the planar isotropic $L_p$ dual Minkowski problem, that is, for any $p,q\in\mathbb{R}$ we obtain the complete classification of the solutions of the following equation: \begin{equation*} u^{1-p}(u_θ^2+u^2)^{\frac{q-2}{2}}(u_{θθ}+u)=1\quad\text{on}\ \mathbb{S}^1. \end{equation*} To establish the classification, we convert the ODE for the solution…
$L_p$ Minkowski problem and Brunn-Minkowski inequality for dual quermassintegrals
本文研究 $L_p$ 对偶 Brunn-Minkowski 理论中的核心问题，包括对偶拟质量积分的 $L_p$ Minkowski 问题和 $L_p$ Brunn-Minkowski 不等式，建立了无对称假设的 $C^0$ 估计，证明了光滑解的唯一性，并研究了原点对称凸体的 $L_p$ Brunn-Minkowski 不等式。
The exact number of nonconstant positive solutions to the planar isotropic $L_p$ dual Minkowski problem
本文确定了平面各向同性$L_p$对偶Minkowski问题非恒定正解的精确个数，通过新参数化与周期单调性分析，给出了完整分类。
In this paper, the $q$-th dual curvature measure is extended to convex functions and the associated Minkowski problem is posed. A special case includes the $q$-th dual curvature measure of convex bodies which defined by Huang, Lutwak, Yang and Zhang. Existence for the functional dual Minkowski problem is showed when $q\leq0$ and the uniqueness part is obtained with some assumptions.
The $L_p$-dual Christoffel-Minkowski problem for the case $p\geq q$
本文研究了 $p\geq q$ 情况下的 $L_p$-对偶 Christoffel-Minkowski 问题，通过结合常数秩定理、先验估计和连续性方法，证明了其严格球面凸解的存在性与唯一性。
Orlicz-type Function Spaces and Generalized Gradient Flows with Degenerate Dissipation Potentials in Non-Reflexive Banach Spaces: Theory and Application
本论文研究非自反Banach空间中广义梯度流强解存在性与Banach值Orlicz空间理论，拓展对偶理论并改进已有结果，结合二者为任意增长率的微分方程分析提供工具，且通过Allen-Cahn-Gurtin型问题弱解存在性验证其有效性。
The $L_p$ dual Christoffel-Minkowski type problem for a class of Hessian quotient equations
本文研究了Hessian商算子σ_k(Λ)/σ_l(Λ)的L_p对偶Christoffel-Minkowski类型问题，利用新发现的'逆凸性'性质，在适当结构假设下建立了满秩定理，并结合先验估计证明了严格球面凸解的存在性和唯一性。
The Weighted Dual Minkowski Problem Under Group Symmetry
本文在群对称框架下研究加权对偶闵可夫斯基问题，推广了原点对称情形的存在性结果，给出了不同q取值下解存在的充要或充分条件，并推广了Henk-Pollehn的集中性质。
On the Brunn-Minkowski inequality for $q$-th dual quermassintegrals with $q>n$
本文针对q>n时q阶对偶曲率积分的Brunn-Minkowski不等式展开研究，证明其对任意凸体及原点对称凸体的失效情形，建立原点对称凸体端点情形与无条件凸体全范围的不等式，并推导对偶曲率测度的唯一性结果。
Timelike Ricci curvature lower bounds via optimal transport for Orlicz-type Lorentzian costs
本文研究了全局超曲时空上的最优运输问题，针对Orlicz型洛伦兹成本函数u∘ℓ，提出u-分离概念并获得强对偶性，通过几何熵凸性刻画时状里奇曲率下界。
Compactness of the $L_p$ dual Minkowski problem in $\mathbb{R}^3$
本文在一般条件下证明了 $S^2$ 上 $L_p$ $q$ 阶对偶闵可夫斯基问题的 $C^0$ 估计，并由此推出近各向同性情形下解的唯一性。
Uniqueness in the near isotropic Lp dual Minkowski problem
本文证明了在 q 接近 n 且 Lp 对偶曲率接近常值时，球面上近各向同性 Lp 对偶 Minkowski 问题解的唯一性，并建立了最优的 C0 估计。
The $L_p$ dual Minkowski problem for unbounded closed convex sets
本文研究C-相容集的$L_p$对偶Minkowski问题，推导无界凸超曲面的新型Monge-Ampère方程，分别在C-确定集族与$(C,p,q)$-接近集族中求解该问题，并得到$p\leq q$时的解唯一性结果。
Nonuniqueness and nonexistence results for the Lp-dual Minkowski problem with supercritical exponents
本文研究了Monge-Ampère型Lp-对偶Minkowski问题，通过推广Chou-Wang恒等式，在特定参数范围内证明了非唯一性及超临界情形下的不存在性结果。
Dafermos' principle and Brenier's duality scheme for defocusing dispersive equations
本文为散焦色散方程建立抽象熵结构，引入矩阵值对偶变分格式，证明其一致性与可解性，并推出Dafermos原理：次解不能比强解更早或更快耗散总熵。
The Lp centro-sectional Minkowski problem
本文引入Lp中心截面闵可夫斯基问题，解决了p>1且q>0时的存在性，讨论了正则性与唯一性，并证明了p较大时的Lp Brunn-Minkowski型不等式。
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