We consider the generalized Wirtinger inequality \[ (\int_{0}^{T} a |u|^q )^{1/q} \le C \biggm(\int_{0}^{T} a^{1-p} |u'|^{p}\biggm)^{1/p}, \] with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge0$, $a\not\equiv0$ and where $u$ is a $T$-periodic function satisfying the constraint \[ \int_{0}^{T} a |u|^{q-2}u =0. \] We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.

2008-03-11math.AP

A sharp weighted Wirtinger inequality

We obtain a sharp estimate for the best constant $C>0$ in the Wirtinger type inequality \[ \int_0^{2π}γ^pw^2\le C\int_0^{2π}γ^qw'^2 \] where $γ$ is bounded above and below away from zero, $w$ is $2π$-periodic and such that $\int_0^{2π}γ^pw=0$, and $p+q\ge0$. Our result generalizes an inequality of Piccinini and Spagnolo.

2005-01-04math.APmath.DS

An optimal Poincaré-Wirtinger inequality in Gauss space

Let $Ω$ be a smooth, convex, unbounded domain of $\R^N$. Denote by $μ_1(Ω)$ the first nontrivial Neumann eigenvalue of the Hermite operator in $Ω$; we prove that $μ_1(Ω) \ge 1$. The result is sharp since equality sign is achieved when $Ω$ is a $N$-dimensional strip. Our estimate can be equivalently viewed as an optimal Poincaré-Wirtinger inequality for functions belonging to the weighted Sobolev space $H^1(Ω,dγ_N)$, where $γ_N$ is the $N$-dimensional Gaussian measure.

2012-10-05math.AP

变分联系Olech-Opial不等式、Wirtinger不等式与Emden-Fowler方程

A Variational link between the Olech-Opial inequality, the Wirtinger inequality, and Emden-Fowler equations

CEA（法国原子能和替代能源委员会）；Université Paris-Saclay（巴黎萨克雷大学）

本文通过混合能量项的积分恒等式，建立Olech-Opial不等式与Wirtinger不等式之间的变分联系，并通过变分问题得到最优常数，其极值满足Emden-Fowler方程。

2026-03-24math.APmath.FA

The equality case in a Poincaré-Wirtinger type inequality

In this paper, generalizing to the non smooth case already existing results, we prove that, for any convex planar set $Ω$, the first non-trivial Neumann eigenvalue $μ_1(Ω)$ of the Hermite operator is greater than or equal to 1. Furthermore, and this is our main result, under some additional assumptions on $Ω$, we show that $μ_1(Ω)=1$ if and only if $Ω$ is any strip. The study of the equality case requires, among other things, an asymptotic analysis of the eigenvalues of the Hermite operator in thin domains.

2014-10-02math.APmath.OCmath.SP

Lipschitz 区域上变指数 Sobolev 空间的模形式 Poincaré-Wirtinger 型不等式

A modular Poincaré-Wirtinger type inequality on Lipschitz domains for Sobolev spaces with variable exponents

该研究针对变指数Sobolev空间中模形式Poincaré-Wirtinger不等式无法退化为经典形式的问题，在有界Lipschitz区域上建立了常指数时可退化为经典形式的广义模不等式，给出构造性证明并补充了最优常数的几何信息。

2024-01-30math.AP

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非局部非线性庞加莱-维尔廷格不等式的对称性结果

Symmetry results for a nonlocal nonlinear Poincaré-Wirtinger inequality

本文研究了$(a,b)\nsubset\boldsymbol{\rm R}$上非局部非线性庞加莱-维尔廷格不等式的最优常数，通过变分方法证明存在临界值$α_C$，确定了极小值函数在$α\nleα_{C}$时为偶且常符号、$α\ngtrα_{C}$时为奇的对称性结果。

2025-08-20math.AP

A generalized mean value inequality for subharmonic functions and applications

We generalize the well-known mean value inequality of subharmonic functions for a slightly more general function class. We also apply this generalized mean value inequality to weighted boundary behavior and nonintegrability questions of subharmonic and superharmonic functions.

2006-11-01math.APmath.CA

Generalized Ponce's inequality

This note provides the generalization of a remarkable inequality by A. C. Ponce whose consequences are essential in several fields, as Characterization of Sobolev Spaces or Nonlocal Modelization.

2019-09-26math.AP

Generalized Polya-Szego inequality and applications to some quasi-linear elliptic problems

We generalize Polya-Szego inequality to integrands depending on $u$ and its gradient. Under minimal additional assumptions, we establish equality cases in this generalized inequality. We also give relevant applications of our study to a class of quasi-linear elliptic equations and systems.

2010-02-18math.AP

Logarithmic mean inequality for generalized trigonometric and hyperbolic functions

In this paper we study the convexity and concavity properties of generalized trigonometric and hyperbolic functions in case of Logarithmic mean.

2014-04-27math.AP

Weighted generalized Korn inequality on John domains

The goal of this work is to show that the generalized Korn inequality that replaces the symmetric part of the differential matrix in the classical Korn inequality by its trace-free part is valid over John domains and weighted Sobolev spaces. The weights considered are nonnegative powers of the distance to the boundary.

2016-12-14math.AP

Harnack's inequality for quasilinear elliptic equations with generalized Orlicz growth

We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.

2020-08-09math.AP

A Generalized Poincare Inequality for a Class of Constant Coefficient Differential Operators

We study first order differential operators with constant coefficients. The main question is under what conditions a generalized Poincaré inequality holds. We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.

2009-10-11math.AP

Generalized Analogs of the Heisenberg Uncertainty Inequality

We investigate locally compact topological groups for which a generalized analogue of Heisenberg uncertainty inequality hold. In particular, it is shown that this inequality holds for $\mathbb{R}^n \times K$ (where $K$ is a separable unimodular locally compact group of type I), Euclidean Motion group and several general classes of nilpotent Lie groups which include thread-like nilpotent Lie groups, $2$-NPC nilpotent Lie groups and several low-dimensional nilpotent Lie groups.

2015-08-11math.AP

Generalized Harnack inequality for semilinear elliptic equations

This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.

2016-07-27math.AP

The weak Harnack inequality for unbounded supersolutions of equations with generalized Orlicz growth

We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. We show that they satisfy the weak Harnack inequality with optimal exponent provided that they belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish the sharpness of our central assumptions.

2020-08-05math.AP

广义Monge-Ampère方程特征值的Lp Brunn-Minkowski不等式

Lp Brunn-Minkowski inequality for the eigenvalue of generalized Monge-Ampère equation

Southeast University（东南大学）；Shanghai Institute for Mathematics and Interdisciplinary Sciences（上海数学与交叉学科研究院）

该研究针对广义Monge-Ampère方程的特征值问题，证明其在凸体的Lp Minkowski和上满足Lp Brunn-Minkowski不等式，完善了相关理论。

2026-09-02math.AP

从Brunn-Minkowski不等式到扭转刚度的一类广义Poincaré型不等式

From the Brunn-Minkowski inequality to a class of generalized Poincaré-type inequalities for torsional rigidity

本文利用Brunn-Minkowski不等式的凹性，在$C^{2}_{+}$类凸体边界上建立了一类关于扭转刚度的广义Poincaré型不等式。

2024-08-29math.AP

A generalized Korn inequality and strong unique continuation for the Reissner-Mindlin plate system

We prove constructive estimates for elastic plates modelled by the Reissner-Mindlin theory and made by general anisotropic material. Namely, we obtain a generalized Korn inequality which allows to derive quantitative stability and global H^2 regularity for the Neumann problem. Moreover, in case of isotropic material, we derive an interior three spheres inequality with optimal exponent from which the strong unique continuation property follows.

2016-10-05math.AP

Generalized Harnack's inequality for nonhomogeneous elliptic equations

This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack's inequality. This paper presents a new type of generalization of the classical Harnack's inequality for such equations. As a corollary we obtain the optimal Harnack type of estimate for p(x)-harmonic functions which quantifies the strong minimum principle.

2019-06-26math.AP

Generalized Hölder's inequality on Morrey spaces

The aim of this paper is to present necessary and sufficient conditions for generalized Hölder's inequality on generalized Morrey spaces. We also obtain similar results on weak Morrey spaces and on generalized weak Morrey spaces. The necessary and sufficient conditions for the generalized Hölder's inequality on these spaces are obtained through estimates for characteristic functions of balls in $\mathbb{R}^d$.

2018-02-17math.APmath.FA

A Generalized Interpolation Inequality and its Application to the Stabilization of Damped Equations

In this paper, we establish a generalized H{ö}lder's or interpolation inequality for weighted spaces in which the weights are non-necessarily homogeneous. We apply it to the stabilization of some damped wave-like evolution equations. This allows obtaining explicit decay rates for smooth solutions for more general classes of damping operators. In particular, for $1-d$ models, we can give an explicit decay estimate for pointwise damping mechanisms supported on any strategic point.

2015-03-24math.AP

Generalized logarithmic Hardy-Littlewood-Sobolev inequality

This paper is devoted to logarithmic Hardy-Littlewood-Sobolev inequalities in the two-dimensional Euclidean space, in presence of an external potential with logarithmic growth. The coupling with the potential introduces a new parameter, with two regimes. The attractive regime reflects the standard logarithmic Hardy-Littlewood-Sobolev inequality. The second regime corresponds to a reverse inequality, with the opposite sign in the convolution term, that allows us to bound the free energy of a drift-diffusion-Poisson system from below. Our method is based on an extension of an entropy method prop…

2019-12-24math.AP

Hardy's Inequality for the fractional powers of Grushin operator

We prove Hardy's inequality for the fractional powers of the generalized sublaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers of generalized sublaplacian.

2017-12-04math.AP

Entire solutions of the generalized Hessian inequality

In this paper, we discuss the more general Hessian inequality $σ_{k}^{\frac{1}{k}}(λ(D_i (A\left(|Du|\right) D_j u)))\geq f(u)$ including the Laplacian, p-Laplacian, mean curvature, Hessian, k-mean curvature operators, and provide a necessary and sufficient condition on the global solvability, which can be regarded as generalized Keller-Osserman conditions.

2022-05-17math.APmath.DG

关于具有广义 Orlicz 增长的椭圆泛函的无界极小元的弱 Harnack 不等式的一个注记

A note on the weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

本文在局部 L^s 可积条件下证明 De Giorgi 类函数的弱 Harnack 不等式，将结果推广至非对数条件下的变指数与双相位广义 Orlicz 增长泛函。

2023-12-07math.AP

A Carleson type inequality for fully nonlinear elliptic equations with non-Lipschitz drift term

This paper concerns the boundary behavior of solutions of certain fully nonlinear equations with a general drift term. We elaborate on the non-homogeneous generalized Harnack inequality proved by the second author in (Julin, ARMA -15), to prove a generalized Carleson estimate. We also prove boundary Hölder continuity and a boundary Harnack type inequality.

2015-08-26math.AP

关于广义的Li-Yau不等式

On generalized Li-Yau inequalities

本文推广了Li-Yau不等式至二阶导数，并建立了四阶导数的Li-Yau型不等式，推导基于热方程的表示公式。

2023-12-17math.AP

基于JKO方案的广义对数Sobolev不等式

Generalized Logarithmic Sobolev Inequality by the JKO Scheme

Universite Sorbonne Paris Nord（索邦巴黎北大学）；IUT de Saint-Denis（圣但尼高等技术学院）；Universite Claude Bernard Lyon 1（里昂第一大学）；ICJ UMR5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet（里昂中央理工学院、里昂国立应用科学学院、让·穆兰大学（ICJ UMR5208，CNRS））

本文基于JKO方案提出离散Bakry-Émery方法，建立了log-concave测度的广义对数Sobolev不等式，并恢复了已知不等式。

2026-02-06math.APmath.OC

On a frequency localized Bernstein inequality and some generalized Poincare-type inequalities

We consider a frequency localized Bernstein inequality for the fractional Laplacian operator which has wide applications in fluid dynamics such as dissipative surface quasi-geostrophic equations. We use a heat flow reformulation and prove the inequality for the full range of parameters and in all dimensions. A crucial observation is that after frequency projection the zero frequency part of the Lévy semigroup does not participate in the inequality and therefore can be freely adjusted. Our proof is based on this idea and a careful perturbation of the Lévy semigroup near the zero frequency which…

2012-12-02math-phmath.APmath.MP

Sufficient and Necessary Conditions for the fractional Gagliardo-Nirenberg Inequalities and applications to Navier-Stokes and generalized boson equations

Sufficient and necessary conditions for the generalized Gagliardo-Nirenberg (GN) inequality in Besov spaces and Triebel-Lizorkin spaces are obtained. Applying the GN inequality, we show that the finite-time blowup solutions have concentration phenomena in critical Lebesgue space L^3. Moreover, we consider the minimizer for a class of variational problem by applying the fractional GN inequality.

2011-01-26math.APmath.FA

均曲率流与古解的广义Harnack不等式

Generalized Harnack Inequality for Mean Curvature Flow and Ancient Solutions

本文通过放松凸性假设，研究均曲率流中广义Harnack不等式及古解的收缩球族特征化。

2025-02-24math.APmath.DG

非负曲率黎曼流形上的sharp p-塞伯利格不等式极值函数的广义存在性

Generalized existence of extremizers for the sharp $p$-Sobolev inequality on Riemannian manifolds with nonnegative curvature

研究非负曲率黎曼流形上sharp p-塞伯利格不等式极值函数的广义存在性，通过非光滑方法和Cheeger能量收敛性分析，揭示极值函数的全局行为。

2025-11-24math.APmath.DGmath.MG