含极限次临界增长项p-Laplace方程的无穷多解

讨论了有界光滑区域上一类p-Laplace方程,非线性项具奇对称性且在无穷远为极限次临界增长．证明了变分泛函在大范围内满足推广的Palais-Smale条件,构造了变分泛函的一列临界值,进而得到了无穷多个弱解的存在性,对应泛函的能量趋于正无穷．所得到的结果推广了次临界增长的情形．     

拟钱性抛物变分不等式

考虑具有多项式增长的拟线性正则抛物变分不等式;利用近似方法和罚技巧,得到了拟正则变分不等式解的存在性和唯一性。     

一类具有变号位势Kirchhoff型方程解的存在性

该文研究了一类带有变号位势非线性项的Kirchhoff型方程的Neumann边值问题.利用变分方法,首先对空间进行分解,证明了该问题的能量泛函满足山路结构;然后证明了能量泛函的(PS)序列有强收敛的子列;最后通过Ekeland变分原理和山路引理,获得了该问题两个非平凡解的存在性.     

Wick型随机组合ZK方程的白噪声泛函解

对变系数组合ZK方程进行白噪声扰动得到的Wick型随机组合ZK方程进行了研究．在Kondratiev分布空间（S）-1中利用白噪声分析,Hermite变换和多项式展开法,得到Wick型随机组合ZK方程的白噪声泛函解和变系数组合ZK方程的精确解．     

一般多项式有界算子的泛函表示定理

该文在圆盘代数A(D)中引入了一个数乘变换, 找到了多项式有界算子的多项式演算与Riesz函数演算之间的联系, 得到了Banach空间X上的一般多项式有界算子的泛函表示定理.     

F-Willmore曲面的间隙现象

对于空间形式中的2维曲面,定义了F-Willmore泛函,此泛函包括经典的Willmore泛函作为特殊情形.F-Willmore泛函的临界点称为F-Willmore曲面.推导了第1变分公式并由此构造了F-Willmore曲面的典型例子.利用自伴算子作用于特殊的实验函数,得到了Simons类积分不等式,讨论了F-Willmore曲面的间隙现象,定出了间隙端点对应的特殊曲面.     

关于弹性力学广义变分原理的等价性定理的研究

利用场论中的不变性原理研究弹性力学广义变分原理的等价性定理,主要目的是研究弹性力学广义变分原理之间的关系;根据弹性力学广义变分原理的泛函在无穷小标度变换下的不变性,证明了这些泛函之间的等价性定理．如果这些泛函具有无穷小标度变换下的不变性,那么只有两类变量是独立的, 应力应变关系是这些泛函必须满足的变分约束条件．所得到的结果再一次证明了钱伟长教授关于所有的弹性力学广义变分原理都是等价的结论．     

Euler-Lagrange方程的判别准则

<正> 给定一个非线性偏微分方程,判别它是否为某个变分问题的Euler-Lagrange方程,这在有限元方法及非线性波理论等许多问题的研究中是一个很重要的问题,因为能由变分原理推出的方程,具有一些独特的性质;而能量泛函的存在性又是有限元等方法的起点.在历史上,最早应用泛函分析方法得到了一般空间中算子位势性的判别     

含多个任意参数的广义变分原理及换元乘子法

弹性力学变分原理的泛函变换可分为三种格式:Ⅰ、放松格式,Ⅱ、增广格式,Ⅲ、等价格式. 根据格式Ⅲ,提出含多个任意参数的广义变分原理及其泛函表示式,其中包括:以位移u为一类泛函变量的多参数广义变分原理;以位移u和应力σ为二类泛函变量的多参数广义变分原理;以位移u和应变ε为二类泛函变量的多参数广义变分原理;以位移u应变ε和应力σ为三类泛函变量的多参数广义变分原理.由这些原理可得出等价泛函一系列新形式,此外,通过参数的合理选择,可构造出一系列有限元模型. 本文还讨论了拉氏乘子法“失效”问题,指出“失效”现象产生的原因,提出乘子法“恢复有效”的作法——换元乘子法.     

Hilbert空间的正交分解及其应用

基于Riesz-表示算子,给出了实Hilbert内积空间按某种连续双线性泛函的正交分解的刻划,应用于鞍点变分问题,获得了解的分离及其强制型于问题.     

Elastic contact problem with Coulomb friction and normal compliance in Orlicz spaces

A static contact problem for inhomogeneous elastic materials is studied with a non-polynomial growth of the elasticity under the Coulomb’s law of dry friction and the normal compliance condition. We demonstrate the results on existence and uniqueness of a solution to an abstract subdifferential inclusion and a variational–hemivariational inequality in the reflexive Orlicz–Sobolev space which are applied to the static elastic frictional problem.     

Adhesive contact problem for viscoplastic materials

We examine a mathematical model that describes a quasistatic adhesive contact between a viscoplastic body and deformable foundation. The material’s behaviour is described by the rate-type constitutive law which involves functions with a non-polynomial growth. The contact is modelled by the normal compliance condition with limited penetration and adhesion, a subdifferential friction condition also depending on adhesion, and the evolution of bonding field is governed by an ordinary differential equation. We present the variational formulation of this problem which is a system of an almost history-dependent variational–hemivariational inequality for the displacement field and an ordinary differential equation for the bonding field. The results on existence and uniqueness of solution to an abstract almost history-dependent inclusion and variational–hemivariational inequality in the reflexive Orlicz–Sobolev space are proved and applied to the adhesive contact problem.     

Orlicz–Young Structures of Young Measures

The paper examines the integration of Young functions applied to Young measures and identifies Orlicz-like structures in the space of Young measures. In particular, a convergence intermediate between the weak convergence of measures and the variational norm is determined; it serves in the completion of the Orlicz space of functions when interpreted as degenerate Young measures. Partial linear operations are defined on Young measures with respect to which the linear operations in the Orlicz space of functions are continuously embedded in the space of Young measures. This leads to a definition of convexity-type structures in the space of Young measures via a limiting procedure. These structures enable applications of Young functions arguments to Young measures. Applications to optimal control and to well posedness of minimization in function spaces with respect to convex functions are provided.     

Dual Orlicz–Brunn–Minkowski theory

In this paper, a dual Orlicz–Brunn–Minkowski theory is presented. An Orlicz radial sum and dual Orlicz mixed volumes are introduced. The dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality are established. The variational formula for the volume with respect to the Orlicz radial sum is proved. The equivalence between the dual Orlicz–Minkowski inequality and the dual Orlicz–Brunn–Minkowski inequality is demonstrated. Orlicz intersection bodies are defined and the Orlicz–Busemann–Petty problem is posed.     

Embedding of Sobolev space in Orlicz space for a domain with irregular boundary

In this paper, we establish the embedding of a weighted Sobolev space in an Orlicz space for a domain with irregular boundary. We find an estimate of the order of growth of the N-function (defining the Orlicz space) and show that, under certain additional constraints on the weights, this estimate is sharp. We also establish the embedding in the space of continuous functions.     

Arbitrarily small weak solutions for a nonlinear eigenvalue problem in Orlicz–Sobolev spaces

Under an appropriate oscillating behavior of the nonlinear term, the existence of a determined open interval of positive parameters for which an eigenvalue non-homogeneous Neumann problem admits infinitely many weak solutions that strongly converges to zero, in an appropriate Orlicz–Sobolev space, is proved. Our approach is based on variational methods. The abstract result of this paper is illustrated by a concrete case.     

The High Resolution Vector Quantization Problem with Orlicz Norm Distortion

We derive a high-resolution formula for the quantization problem under Orlicz norm distortion. In this setting, the optimal point density solves a variational problem which comprises a function g:ℝ₊→[0,∞) characterizing the quantization complexity of the underlying Orlicz space. Moreover, asymptotically optimal codebooks induce a tight sequence of empirical measures. The set of possible accumulation points is characterized, and in most cases it consists of a single element. In that case, we find convergence as in the classical setting.     

An existence result for a nonhomogeneous problem in R2 related to nonlinear Hencky-type materials

This paper investigates a nonlinear and non-homogeneous system of partial differential equations. The motivation comes from the fact that in a particular case the problem discussed here can be used in modeling the behavior of nonlinear Hencky-type materials. The main result of the paper establishes the existence of a nontrivial solution in an adequate functional space of Orlicz–Sobolev type by using Schauder’s fixed point theorem combined with adequate variational techniques.     

Orlicz序列空间的一致单调系数及应用

本文给出了Ｏｒｌｉｃｚ序列空间一致单调系数数值，同时给出了Ｏｒｌｉｃｚ序列空间具有一致单调性的条件，进而讨论具有一致单调性的Ｏｒｌｉｃｚ序列空间中的最佳逼近算子的一些特征。