The Orlicz Minkowski problem for logarithmic capacity seeks to determine the necessary and sufficient conditions for a given finite Borel measure,such that it is the Orlicz logarithmic capacitary measure of a convex b...The Orlicz Minkowski problem for logarithmic capacity seeks to determine the necessary and sufficient conditions for a given finite Borel measure,such that it is the Orlicz logarithmic capacitary measure of a convex body.The Orlicz Minkowski problem for loga-rithmic capacity includes the Minkowski problem for logarithmic capacity and the Lp Minkowski problem for logarithmic capacity as special cases.The discrete case has been solved by the researchers.In this paper,we solve the Orlicz Minkowski problem for logarithmic capacity with respect to general Borel measures by applying an approximation scheme.展开更多
In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvatu...In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for p>−n with p≠0.展开更多
Chord measures are newly discovered translation-invariant geometric measures of convex bodies in Rn by Lutwak-Xi-Yang-Zhang(Communications on Pure and Applied Mathematics,2024),which is an extension of the surface are...Chord measures are newly discovered translation-invariant geometric measures of convex bodies in Rn by Lutwak-Xi-Yang-Zhang(Communications on Pure and Applied Mathematics,2024),which is an extension of the surface area measure.The Minkowski problems for chord measures was considered by Lutwak-Xi-Yang-Zhang.In this paper,we use variational method to solve the even Orlicz chord Minkowski problem.The obtained results are an extension of the even Orlicz Minkowski problem from Haberl-Lutwak-Yang-Zhang(Advances in Mathematics,2010).展开更多
In this paper,we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case.Meanwhile,the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for ...In this paper,we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case.Meanwhile,the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.展开更多
In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the...In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].展开更多
In this paper,we provide a sufficient condition,in the case of 0<p<1,for the existence of solutions to the general L_(p) Minkowski problem for polytopes.
The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article com...The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article completely solves the case of discrete measures whose support sets are in general position.展开更多
In this paper,it is proved that the weak convergence of the L_(p) Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p≥1.Moreover,continuity of the s...In this paper,it is proved that the weak convergence of the L_(p) Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p≥1.Moreover,continuity of the solution to the L_(p) Gaussian Minkowski problem with respect to p is obtained.展开更多
In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then ...In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then there is a smooth and strictly convex body solving the planar L_(p) dual Minkowski problem.展开更多
This paper concerns the even L_(p)Gaussian Minkowski problem in n-dimensional Euclidean space R^(n).The existence of the solution to the even L_(p)Guassian Minkowski problem for p>n is obtained.
The centroafhine Minkowski problem is studied, which is the critical case of the L_p-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions.In this pap...The centroafhine Minkowski problem is studied, which is the critical case of the L_p-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions.In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.展开更多
This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R^2 implies the convergence of ...This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R^2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.展开更多
Schuster introduced the notion of Blaschke-Minkowski homomorphisms and first considered Busemann-Petty type problems. In this paper, we study the Busemann-Petty type problems for the geominimal surface area with respe...Schuster introduced the notion of Blaschke-Minkowski homomorphisms and first considered Busemann-Petty type problems. In this paper, we study the Busemann-Petty type problems for the geominimal surface area with respect to Blaschke-Minkowski homomorphisms.展开更多
In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyp...In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n =2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem; while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.展开更多
基金Supported by Postgraduate Scientific Research Innovation Project of Hunan Province(CX20231033)Science and Technology Research Project of Jiangxi Provincial Education Department(GJJ210815)+2 种基金Jiangxi Provincial Natural Science Foundation(20232BAB201005)the National Natural Science Founda-tion of China(12461010,12161043)the Scientific Research Fund of Hunan Provincial Education Department(24A0338)。
文摘The Orlicz Minkowski problem for logarithmic capacity seeks to determine the necessary and sufficient conditions for a given finite Borel measure,such that it is the Orlicz logarithmic capacitary measure of a convex body.The Orlicz Minkowski problem for loga-rithmic capacity includes the Minkowski problem for logarithmic capacity and the Lp Minkowski problem for logarithmic capacity as special cases.The discrete case has been solved by the researchers.In this paper,we solve the Orlicz Minkowski problem for logarithmic capacity with respect to general Borel measures by applying an approximation scheme.
基金supported by the National Natural Science Foundation of China(12171144,12231006,12122106).
文摘In this paper,the L_(p)chord Minkowski problem is concerned.Based on the results shown in[20],we obtain a new existence result of solutions to this problem in terms of smooth measures by using a nonlocal Gauss curvature flow for p>−n with p≠0.
基金Supported by the National Natural Science Foundation of China(12071277,12071334)。
文摘Chord measures are newly discovered translation-invariant geometric measures of convex bodies in Rn by Lutwak-Xi-Yang-Zhang(Communications on Pure and Applied Mathematics,2024),which is an extension of the surface area measure.The Minkowski problems for chord measures was considered by Lutwak-Xi-Yang-Zhang.In this paper,we use variational method to solve the even Orlicz chord Minkowski problem.The obtained results are an extension of the even Orlicz Minkowski problem from Haberl-Lutwak-Yang-Zhang(Advances in Mathematics,2010).
基金The authors were supported by NSFC(11771132)Hunan Science and Technology Project(2018JJ1004).
文摘In this paper,we prove the uniqueness of the Lp Minkowski problem for q-torsional rigidity with p>1 and q>1 in smooth case.Meanwhile,the Lp Brunn-Minkowski inequality and the Lp Hadamard variational formula for q-torsional rigidity are established.
基金Supported by National Natural Science Foundation of China(12171260).
文摘In this paper,we consider generalized Christo®el-Minkowski problems as followsσ_(k)(u_(ij)+uδ_(ij))/σ_(l)(u_(ij)+uδ_(ij))=|u^(p-1)f(x),x∈S^(n),where 0≤l≤k≤n,p-1>0 and f is positive,and we establish the weighted gradient estimate and uniform C^(0)estimate for the positive convex even solutions,which is a generalization of Guan-Xia[1]and Guan[2].
基金Supported by the National Natural Science Foundation of China(Grant Nos.12371060,12401069)the Scientific Research Startup Fund for PhD Recipients(Grant No.BS25006)+1 种基金the Recruitment Program for Young Professionals of Chinathe Fundamental Research Funds for the Central Universities(Grant No.GK202307001)。
文摘In this paper,we provide a sufficient condition,in the case of 0<p<1,for the existence of solutions to the general L_(p) Minkowski problem for polytopes.
文摘The logarithmic capacitary Minkowski problem asks for necessary and sufficient conditions on a finite Borel measure on the unit sphere so that it is the logarithmic capacitary measure of a convex body.This article completely solves the case of discrete measures whose support sets are in general position.
基金Supported by China Postdoctoral Science Foundation(Gratn No.2020M682222)Natural Science Foundation of Shandong Province(Grant Nos.ZR2020QA003,ZR2020QA004)。
文摘In this paper,it is proved that the weak convergence of the L_(p) Gaussian surface area measures implies the convergence of the corresponding convex bodies in the Hausdorff metric for p≥1.Moreover,continuity of the solution to the L_(p) Gaussian Minkowski problem with respect to p is obtained.
基金supported by National Natural Science Foundation of China(Grant Nos.11971424 and 11571304)。
文摘In this paper we study the L_(p) dual Minkowski problem for the case p<0<q.We prove for any positive smooth function f on S^(1),there exists an F:R^(+)→R^(-),such that if F(q)<p<0 or 0<q<-F(-p)then there is a smooth and strictly convex body solving the planar L_(p) dual Minkowski problem.
基金supported by the National Natural Science Foundation of China(No.12301066)China Postdoctoral Science Foundation(No.2020M682222)the Natural Science Foundation of Shandong Province(No.ZR2020QA003)。
文摘This paper concerns the even L_(p)Gaussian Minkowski problem in n-dimensional Euclidean space R^(n).The existence of the solution to the even L_(p)Guassian Minkowski problem for p>n is obtained.
基金supported by National Natural Science Foundation of China (Grant No 11401527)
文摘The centroafhine Minkowski problem is studied, which is the critical case of the L_p-Minkowski problem. It admits a variational structure that plays an important role in studying the existence of solutions.In this paper, we find that there is generally no maximizer of the corresponding functional for the centroaffine Minkowski problem.
基金supported by National Natural Science Foundation of China (Grant No. 11671325)
文摘This paper investigates continuity of the solution to the even logarithmic Minkowski problem in the plane. It is shown that the weak convergence of a sequence of cone-volume measures in R^2 implies the convergence of the sequence of the corresponding origin-symmetric convex bodies in the Hausdorff metric.
基金Supported by the National Natural Science Foundation of China(11371224)Innovation Foundation of Graduate Student of China Three Gorges University(2018SSPY136)
文摘Schuster introduced the notion of Blaschke-Minkowski homomorphisms and first considered Busemann-Petty type problems. In this paper, we study the Busemann-Petty type problems for the geominimal surface area with respect to Blaschke-Minkowski homomorphisms.
基金Supported by the NNSF of China(11271323,91330105)the Zhejiang Provincial Natural Science Foundation of China(LZ13A010002)
文摘In this paper, we investigate the solutions of the hyperbolic Yamabe problem for the （1 ＋ n）-dimensional Minkowski space-time. More precisely speaking, for the case of n = 1, we derive a general solution of the hyperbolic Yamabe problem; for the case of n =2, 3, we study the global existence and blowup phenomena of smooth solutions of the hyperbolic Yamabe problem; while for general multi-dimensional case n ≥ 2, we discuss the global existence and non-existence for a kind of exact solutions of the hyperbolic Yamabe problem.
