一般形式Gerstewitz泛函的若干性质及应用

在拓扑向量空间的适当假设下,讨论了非线性标量化函数Gerstewitz泛函的若干性质,包括Gerstewitz泛函的非凸分离性质.此外,文中还建立了一种极小点集的子集与Gerstewitz泛函标量化问题的极小解集之间的对应关系.     

具非S1值边界条件的Ginzburg-Landau 泛函的径向极小元

作者研究了具非S1值边界条件的Ginzburg-Landau泛函的径向极小元的唯—性, 收敛性.此外,作者还讨论了径向极小元的收敛速度.     

一种新的广义次梯度及其性质

该文提出了一种新的广义次梯度,讨论了与Tangent锥相关的一些基本性质,且在部分性质基础上得出了含不等式与任意集约束下非凸非可微标量最优化问题的k阶严格局部极小点存在的必要与充分条件以及无约束非凸非可微标量最优化问题局部极小点存在的必要条件.     

Banach空间上凸泛函的某些对偶性质

本语文对Ｂａｎａｃｈ空间上的凸泛函建立了若干对偶性质,通过对偶的手段,建立了非常简洁的凸泛函极小化序列弱收敛的特征定理。     

一类约束变分问题极小元的存在性及其集中行为

该文主要讨论了一类带有调和位势p-Laplacian方程特征值问题对应的变分泛函极小元的存在性与非存在性,并且使用能量估计的方法分析了当方程中相关参数逼近其临界值时极小元的集中行为.     

关于含非凸约束的变分问题

本文分别讨论泛函在一般的非凸几何约束和非凸图约束下的极小化问题，其中Ｍ为ＲＮ中的任一光滑开子集或ＲＮ的一个开子流形．应用线性化和截断法等，得到了广义有界极小的存在性．     

R~N中分数阶Choquard方程的约束极小元

研究R~N中一类非线性分数阶Choquard方程在指定L2范数下解的存在性.首先,证明了当p∈((N+α)/N,(N+α)/(N-2s))时,具有最佳常数的内插不等式;其次,对于p=(N+α+2s)/N,讨论了限制在L~2范数下方程的极小化问题,得到了能量泛函的限制极小值点的存在性.     

轨道结构的层次与拓扑半共轭

对离散系统提出轨道结构存在3个层次的观念,引进拟弱几乎周期点的概念和关于拓扑半共轭的极小覆盖的概念,讨论了轨道结构3个层次与点的回复性之间的关系以及某些性质在拓扑半共轭下的保持问题.     

极大极小问题的广义投影算法

研究了无约束极大极小问题.通过引入一个可微的辅助函数,利用广义投影技术产生下降搜索方向,结合Armjio非精确线搜索建立了一个广义梯度投影算法.在初始点任意的条件下,证明了算法的全局收敛性.     

不连续丰满映照及其不动点计算

用积分型求总极值的方法,我们可以求出不连续的丰满函数的总极小值和总极小点集.在这篇文章中,我们引进丰满映照的概念,并讨论它的基本性质,把求丰满映照不动点的问题化为求丰满函数的总体极小点集问题,从而可以用积分型求总极小方法求出丰满映照的不动点.实算表明,这个方法是很有效的.     

On the Existence of Solutions of Two Optimization Problems

In this paper, we prove the existence of solutions for the minimization problem of the shell weight for a given minimal frequency of the shell vibrations as well as for the maximization problem of the minimal frequency for a given shell weight. We consider an optimal control problem governed by an eigenvalue problem for a system of differential equations with variable coefficients. The form of the shell is considered as a control. Some of the coefficients are non-measurable. Earlier, we introduced certain special weighted functional spaces. By using these spaces, we establish the continuity of the considered minimal frequency functional and obtain the existence of solutions of both optimal control problems. At the end, we prove the Lipschitz continuity of the eigenvalue problem.     

Some Properties of General Minimization Problems with Constraints

The paper studies the existence of solutions and necessary conditions of optimality for a general minimization problem with constraints. Although we focus mainly on the case where the cost functional is locally Lipschitz, a general Palais–Smale condition is proposed and some of its properties are studied. Applications to an optimal control problem and a Lagrange multiplier rule are also given.     

On a double degenerate fourth-order parabolic equation

The existence of solutions to a fourth-order p-Laplacian equation with boundary degeneracy is studied. For the purpose of solving the corresponding non-degenerate (with respect to the coefficient of fourth-order term) regularized problem, a fourth-order semi-discrete elliptic problem with homogeneous boundary conditions is established and its existence and uniqueness are obtained by the functional minimization method. It follows that the approximate solutions of the non-degenerate parabolic problem are constructed and the corresponding existence and uniqueness are discovered by a limit procedure from the energy estimation method and a compactness argument. Finally, the existence and regularity of solutions for the problem with boundary degeneracy is obtained by using a regularization parameter vanishing limit.     

Qualitative Properties of Eigenvectors Related to Multivalued Operators and some Existence Results

In this work, we study a class of nonlinear eigenvalue problems related to fully discontinuous operators. In particular, we prove the existence of a critical point for two distinct problems. Connected with this problem, we also study a minimization problem with constraint, and we investigate the existence of solutions for a resonant case near zero. Moreover, we give some estimates and qualitative properties of solutions by using the relative rearrangement theory.     

A variational method using fractional order Hilbert spaces for tomographic reconstruction of blurred and noised binary images

We provide in this article a refined functional analysis of the Radon operator restricted to axisymmetric functions, and show that it enjoys strong regularity properties in fractional order Hilbert spaces. This study is motivated by a problem of tomographic reconstruction of binary axially symmetric objects, for which we have available one single blurred and noised snapshot. We propose a variational approach to handle this problem, consisting in solving a minimization problem settled in adapted fractional order Hilbert spaces. We show the existence of solutions, and then derive first order necessary conditions for optimality in the form of optimality systems.     

Existence of solutions of non-autonomous second order functional differential equations with infinite delay

In this paper the existence of solutions of a non-autonomous abstract retarded functional differential equation of second order with infinite delay is considered. Assuming the existence of an evolution operator corresponding to the associate abstract Cauchy problem of second order, we establish the existence of mild solutions of the functional equation. Furthermore, we study the existence of classical solutions of the abstract Cauchy problem of second order and we apply these results to establish the existence of classical solutions of the functional equation. Finally, we apply our results to study the existence of solutions of the non-autonomous wave equation with delay.     

Existence of charged vortices in a Maxwell–Chern–Simons model

In this paper, we prove the existence of charged vortex solitons in a Maxwell–Chern–Simons model. We establish the main existence theorem by a constrained minimization method applied on an indefinite action functional which is induced from the original field-theoretical Lagrangian. We also show that the solutions obtained are smooth.     

Three nontrivial solutions for a nonlinear anisotropic nonlocal equation

In this article, we establish the existence of a sign-changing solution and two sign-constant solutions for nonlinear nonlocal problem involving the BO-ZK operator on bounded domain. Our main tool is constrained minimization on appropriate Nehari manifolds.     

A Model for the Evolution of Laminates

We study the time evolution of a generalized standard material in elastoplasticity. Of our particular interest are the formation and the evolution of microstructure. Our aim is to prove the existence of solutions. This is a challenging task, since the presence of microstructure comes along with a lack of convexity and, hence, compactness arguments cannot be applied to prove the existence of solutions. In order to overcome this problem, we will incorporate information on the microstructure into the internal variable, which is still compatible with the notion of generalized standard materials. More precisely, we shall allow such forms of microstructure that are given by simple laminates. We will consider a model for the evolution of these laminates and we will state a result on the existence of solutions to the time-incremental minimization problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On a class of elliptic problems with indefinite nonlinearities

In this paper we apply variational and sub-supersolution methods to study the existence and multiplicity of nonnegative solutions for a class of indefinite semilinear elliptic problems that depend on a parameter. The results on the existence of solutions do not impose any growth condition at infinity on the term which depends on the parameter. To derive such results, first we find a positive supersolution by solving an auxiliary problem. Then we use a truncation argument and a global minimization method. The main hypothesis for the existence of two nonzero solutions is that the indefinite term is the product of a weight function, having a thick zero set, and a nonlinear function which satisfies the Ambrosetti–Rabinowitz superlinear condition. Results for some corresponding indefinite problems are also established.