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An Interpolation of Hardy inequality and moser–trudinger inequality on Riemannian Manifolds with Negative Curvature

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Acta Mathematica Sinica,English Series 2016年第7期32卷 856-866页

作者： Yan Qing DONG Qiao Hua YANG School of Mathematics and Statistics Wuhan University

Let M be a complete, simply connected Riemannian manifold with negative curvature. We obtain an interpolation of Hardy inequality and moser-trudinger inequality on M. Furthermore, the constant we obtain is sharp.

关键词： moser-trudinger inequality Hardy inequality Riemannian manifold negative curvature

Remarks on the Extremal Functions for the moser-trudinger inequality

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Acta Mathematica Sinica,English Series 2006年第2期22卷 545-550页

作者： Yu Xiang LI Department of Mathematical Science Tsinghua University

We will show in this paper that if A is very close to 1, thenI（M,λ,m） =supu∈H0^1,n（m）,∫m｜△↓u｜^ndV=1∫Ω（e^αn｜u｜^n/（n-1）-λm∑k=1｜αnun/（n-1）｜k/k!）dVcan be attained, where M is a compact-manifold with boundary. This result gives a counter-example to the conjecture of de Figueiredo and Ruf in their paper titled ＂On an inequality by trudinger and moser and related elliptic equations＂（Comm. Pure. Appl. Math., 55, 135-152, 2002）.

关键词： moser-trudinger inequality extremal function

Singular moser-trudinger inequality Involving L^(n) Norm in the Entire Euclidean Space

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Communications in Mathematics and Statistics 2021年第4期9卷 467-501页

作者： Changliang Zhou Chunqin Zhou School of Science East China University of TechnologyNanchangPeople’s Republic of China School of Mathematical Sciences MOE-LSCShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China

In this paper,we investigate a singular moser-trudinger inequality involving L^(n) norm in the entire Euclidean *** blow-up procedures are used for the maximizing *** we obtain the existence of extremal functions for this singular geometric inequality in whole *** general,W^(1,n)(R^(n))→L^(q)(R^(n))is a continuous embedding but not *** in our case we can prove that W^(1,n)(R^(n))→L^(n)(R^(n))is a compact *** the compact embedding W^(1,n)(R^(n))→Lq(R^(n),|x|^(−s)dx)for all q≥n and 0

关键词： moser-trudinger inequality Blow-up analysis Existence of extremal functions

Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Science China Mathematics 2024年第3期67卷 557-576页

作者： Jiaxiang Wang Bin Zhou Beijing International Center for Mathematical Research Peking UniversityBeijing 100871China School of Mathematical Sciences Peking UniversityBeijing 100871China

In this paper, we study the relations between trace inequalities(Sobolev and moser-trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Amp`ere equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

关键词： complex Monge-Ampère equations plurisubharmonic functions Sobolev inequality moser-trudinger inequality

Extremal functions for the moser-trudinger inequalities on compact Riemannian manifolds

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Science China Mathematics 2005年第5期48卷 618-648页

作者： LI Yuxiang Department of Mathematical Sciences, Tsinghua University, Beijing 100084, China Department of Mathematical Sciences Tsinghua University Beijing China

Let (M, g) be a compact Riemannian manifold without boundary, and (N, g) a compact Riemannian manifold with boundary. We will prove in this paper that the ∫MudVg=o,sup∫M|(△↓)u|ndVg=1∫Meαn|u|n/n-1dVg,∫M(|(△↓)|n+|u|n)dVg=1∫Meαn|u|n/n-1dVg,and u|(e)N=0,∫Msup|(△↓)u|dVgN=1∫Neαn|u|n/n-1dVgn can be attained. Our proof uses the blow-up analysis.

关键词： moser-trudinger inequality extremal function n-Lapalace Green function.

A Priori Estimate for the Complex Monge-Ampère Equation

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Peking Mathematical Journal 2021年第1期4卷 143-157页

作者： Jiaxiang Wang Xu-Jia Wang Bin Zhou School of Mathematical Sciences Zhejiang UniversityHangzhou 310027China Centre for Mathematics and its Applications The Australian National UniversityCanberraACT 2601Australia School of Mathematical Sciences Peking UniversityBeijing 100871China

In this paper,we use the Sobolev type inequality in Wang et al.(moser-trudinger inequality for the complex Monge-Ampère equation,arXiv:2003.06056 v1(2020))to establish the uniform estimate and the Hölder continuity for solutions to the com-plex Monge-Ampère equation with the right-hand side in Lp for any given p>*** proof uses various PDE techniques but not the pluripotential theory.

关键词： Complex Monge-Ampère equation moser-trudinger inequality Regularity

A Multiplicity Result for a Singular and Nonhomogeneous Elliptic Problem in R^n

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Journal of Partial Differential Equations 2012年第1期25卷 90-102页

作者： ZHAO Liang School of Mathematical Sciences Beijing Normal UniversityBeijing 100875China and Key Laboratory of Mathematics and Complex SystemsMinistry of EducationBeijing Normal UniversityBeijing 100875China

We establish sufficient conditions under which the quasilinear equation [GRAPHICS] has at least two nontrivial weak solutions in W"(Rn) when e> 0 is small enough, 0 0 and h 0 belongs to the dual space of W"(Rn).

关键词： moser-trudinger inequality exponential growth COMPACT RIEMANNIAN MANIFOLD CRITICAL GROWTH EQUATIONS