一类抛物型H-半变分不等式

研究一类拟线性抛物型H-半变分不等式,即研究具有非凸、非光滑泛函的抛物型变分不等式。这类问题的研究来自力学。利用Clarke广义梯度和伪单调算子理论,证明了一类拟线性抛物型H-半变分不等式解的存在性。     

Riemann-Liouville分数阶半线性发展型H-半变分不等式的可解性和最优控制

本文研究了Hilbert空间中半线性Riemann-Liouville分数阶发展型H-半变分不等式的可解性和最优控制.首先,利用不动点理论和Clarke广义次微分性质得到半线性Riemann-Liouville分数阶发展型H-半变分不等式解的存在性.其次,在一般假设条件下证明系统的最优控制存在性.最后,给出一个例子来验证本文的主要结果.     

一类非线性所变分不等式

本文利用伪单调算子理论研究如下变分不等式问题，求ｘ∈Ｍ，使得并将所得结果应用于拟线性椭圆型边值问题的求解。     

一个退化的变分不等式的数值解

1.引言 稳态渗流自由边界问题大部可化为椭圆型变分不等式或拟变分不等式,其数值解已为不少作者所研究(参看[1],[2]及其文献)。某些轴对称问题则可化为另一类变分不等式一退化的椭圆型变分不等式。[3],[4]和[5]即用此法研究轴对称渗流井(水     

关于一类半线性高阶障碍问题解的存在性与正则性

本文考虑一类带互补边界条件的四阶半线性椭圆型变分不等式,应用临界点理论,证明了当半线性项具有超线性增长时解的存在性,并利用L~p估计得到了解的正则性.     

抛物型H-半变分不等式的收敛性

本文讨论的对象是非线性抛物型H-半变分不等式，使用文献[4]中抛物型G收敛的定义来研究抛物型H-半变分不等式解的收敛性行为。     

隐变分不等式研究的半序方法*

本文利用半序方法在Hausdorff拓扑线性空间中研究了一类一般形式的单调型隐变分不等式解的存在性问题。作为应用的例子,在文末我们应用所得结果,讨论了Nash平衡问题及半线性椭圆型方程解的存在性问题。     

H-半变分不等式逼近解的强收敛性

在Banach空间中,研究H-半变分不等式不适定问题的正则化方法. 假定所研究的H-半变分不等式是可解的,利用Browder-Tikhonov正则化方法构造出强收敛的逼近步骤, 所得出的结论是前人结论的推广和延拓.     

H-空间中上半连续集值映象一个新的不动点定理及其应用

本文在H-空间中建立了上半连续集值映象的一个新的不动点定理,作为应用,得到了两个新的拟变分不等式的解的存在性定理。     

星形集约束微分变分不等式解的存在性北大核心CSCD

由微分方程和变分不等式构成的微分变分不等式是非线性分析及其应用领域中的一类非常重要的问题,吸引了不少学者的极大关注和探索.本文研究一类具有非凸约束的微分变分不等式新问题的解的存在性.该类问题中的变分不等式的约束集是关于某一球的星形集,使得可以利用距离函数的广义Clarke次微分的不连续性质.我们通过多值伪单调算子的满射定理,H-半变分不等式逼近和参数不需要趋于零的罚方法证明解的存在性,并举例说明主要结果在具有非凸约束的抛物型初值问题中的应用.     

Well-posedness of a general class of elliptic mixed hemivariational–variational inequalities

In this paper, well-posedness of a general class of elliptic mixed hemivariational–variational inequalities is studied. This general class includes several classes of the previously studied elliptic mixed hemivariational–variational inequalities as special cases. Moreover, our approach of the well-posedness analysis is easily accessible, unlike those in the published papers on elliptic mixed hemivariational–variational inequalities so far. First, prior theoretical results are recalled for a class of elliptic mixed hemivariational–variational inequalities featured by the presence of a potential operator. Then the well-posedness results are extended through a Banach fixed-point argument to the same class of inequalities without the potential operator assumption. The well-posedness results are further extended to a more general class of elliptic mixed hemivariational–variational inequalities through another application of the Banach fixed-point argument. The theoretical results are illustrated in the study of a contact problem. For comparison, the contact problem is studied both as an elliptic mixed hemivariational–variational inequality and as an elliptic variational–hemivariational inequality.     

Inverse coefficient problems for nonlinear elliptic variational inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic variational inequalities. The unknown coefficient of elliptic variational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic variational inequalities is unique solvable for the given class of coefficients. The existence of quasisolutions of the inverse problems is obtained.     

Inverse Coefficient Problems for Elliptic Hemivariational Inequalities

This paper is devoted to a class of inverse coefficient problems for nonlinear elliptic hemivariational inequalities. The unknown coefficient of elliptic hemivariational inequalities depends on the gradient of the solution and belongs to a set of admissible coefficients. It is shown that the nonlinear elliptic hemivariational inequalities are uniquely solvable for the given class of coefficients. The result of existence of quasisolutions of the inverse problems is obtained.     

Positive weak solutions of semilinear second order elliptic inequalities via variational inequalities

Existence, uniqueness and convergence of approximants of positive weak solutions for semilinear second order elliptic inequalities are obtained. The nonlinearities involved in these inequalities satisfy suitable upper or lower bound conditions or monotonicity conditions. The lower bound conditions are allowed to contain the critical Sobolev exponents. The methodology is to establish variational inequality principles for demicontinuous pseudo-contractive maps in Hilbert spaces by considering convergence of approximants and apply them to the corresponding variational inequalities arising from the semilinear second order elliptic inequalities. Examples on the existence, uniqueness and convergence of approximants of positive weak solutions of the semilinear second order elliptic inequalities are given.     

Minimization principles for elliptic hemivariational inequalities

In this paper, we explore conditions under which certain elliptic hemivariational inequalities permit equivalent minimization principles. It is shown that for an elliptic variational–hemivariational inequality, under the usual assumptions that guarantee the solution existence and uniqueness, if an additional condition is satisfied, the solution of the variational–hemivariational inequality is also the minimizer of a corresponding energy functional. Then, two variants of the equivalence result are given, that are more convenient to use for applications in contact mechanics and in numerical analysis of the variational–hemivariational inequality. When the convex terms are dropped, the results on the elliptic variational–hemivariational inequalities are reduced to that on “pure” elliptic hemivariational inequalities. Finally, two representative examples from contact mechanics are discussed to illustrate application of the theoretical results.     

Confluent and Open Retractible Continua

In this paper we deal with quasilinear elliptic hemivariational inequalities of higher order as generalizations of elliptic variational inequalities of higher order to nonconvex functionals. This extension is strongly motivated by various problems in mechanics. Using the notion of the generalized gradient of Clarke, existence results of solutions have been obtained.     

Finite Element Approximation of Hemivariational Inequalities

The paper deals with a finite element approximation of elliptic and parabolic variational inequalities. Elliptic hemivariational inequalities are equivalently expressed as a system consisting of one equation and one inclusion for a couple of unknowns, namely a primal variable u and an element belonging to a multivalued mapping at u. Both components of the solution are approximated independently each other by a finite element method. Parabolic inequalities are transformed into a system of elliptic ones by using an appropriate time discretization. A numerical experiment is realized by using non-smooth optimization methods.     

Pointwise symmetrization inequalities for Sobolev functions and applications

We develop a technique to obtain new symmetrization inequalities that provide a unified framework to study Sobolev inequalities, concentration inequalities and sharp integrability of solutions of elliptic equations.     

Lower bounds for generalized eigenvalues of the quasilinear systems

In this paper, we establish several new Lyapunov-type inequalities for two classes of one-dimensional quasilinear elliptic systems of resonant type, which generalize or improve all related existing ones. Then we use the Lyapunov-type inequalities obtained in this paper to derive a better lower bound for the generalized eigenvalues of the one-dimensional quasilinear elliptic system with the Dirichlet boundary conditions.     

Nonexistence of sign-changing solutions for some elliptic and parabolic inequalities

We establish nonexistence of nontrivial solutions (including sign-changing ones) for some partial differential inequalities of elliptic and parabolic type containing nonlinear terms that depend on the positive and negative part of the sought function in different ways. Systems of elliptic inequalities with similar structure are also considered. The proofs are based on the test function method.