Singular Functions and Characterizations of Field Concentrations: a Survey

In the presence of closely located inclusions of the extreme material property, the physical fields, such as the electric field and the stress tensor, may be concentrated and arbitrarily large in the narrow region between two inclusions. Recently there has been significant progress on the quantitative characterization of the field concentration in the contexts of electrostatics(Laplace equation), linear elasticity(Lamé system), and viscous flow(Stokes system). This paper is to review such progress in a coherent way.     

各向异性材料界面共线刚性线夹杂的反平面问题

研究两种各向异性材料焊接界面含共线刚性线夹杂的反平面问题,导出了一般问题的公式和几个典型问题的封闭形式解,求出了刚性线尖端的应力分布.从文中解答的特殊情况,直接导出各向同性材料界面与均匀各向异性介质中相应问题的公式与结果,并与有关文献相一致.     

重复频率脉冲激光作用下膜内包裹物对损伤阈值的影响

利用介质薄膜中包裹物的热理论模型,结合S.Papernov 等利用电子束蒸发技术在熔融石英上沉积含Au包裹物的HfO2薄膜实验,得出Au的吸收截面。以包裹物Au为例,计算了脉冲激光作用下不同包裹物半径对损伤阈值的影响,分析了重复率脉冲激光作用下薄膜损伤阈值的变化及重复频率与激光损伤阈值的关系。结果表明：随着包裹物半径的增加,激光损伤阈值先减小,接着增加而后再减小。激光损伤阈值与脉冲宽度的0.4次方成正比。随着脉冲重复频率的增加,激光损伤阈值单调下降,产生损伤所需的最小脉冲数则单调上升。     

Contractive multifunctions, fixed point inclusions and iterated multifunction systems

We study the properties of multifunction operators that are contractive in the Covitz-Nadler sense. In this situation, such operators T possess fixed points satisfying the relation x∈Tx. We introduce an iterative method involving projections that guarantees convergence from any starting point x₀∈X to a point x∈XT, the set of all fixed points of a multifunction operator T. We also prove a continuity result for fixed point sets XT as well as a “generalized collage theorem” for contractive multifunctions. These results can then be used to solve inverse problems involving contractive multifunctions. Two applications of contractive multifunctions are introduced: (i) integral inclusions and (ii) iterated multifunction systems.     

非凸集值映射的包含切性及应用

本文在一般的Ｂａｎａｃｈ空间Ｘ中研究从非空闭集ＫＸ到Ｘ的非凸集值映射Ｆ的包含切性问题．得到的结果定理３．１把有关的结论推广到非光滑空间，定理３．３则将有限维空间的正则性定理推广到任意的Ｂａｎａｃｈ空间．作为结果的应用，我们证明了无穷维非凸微分包含和非凸控制系统生存解的存在性，且给出了一个方便的等价切性条件．     

Liapunov-type inequality for delay-differential equations of third order

A Liapunov-type inequality for a class of third order delay-differential equations is derived.     

Non smooth dynamics of mechanical systems with history term

Models involving maximal monotone term and history (delay) term are considered in a mathematical and numerical point of view.     

Systems of Inclusions Involving Fredholm Operators of Nonnegative Index and Nonconvex-Valued Maps

We study the existence of solutions to a system of two inclusions involving Fredholm operators of nonnegative (Fredholm) index and the so-called c-admissible maps: upper semicontinuous maps with values being continuous images of cell-like sets. The presented approach resembles the substitution technique, i.e. applies the so-called solution map. The appearance of the ‘dimension defect’ reflecting the nonnegativity of Fredholm indices requires the use of algebraic tools based on the cohomotopy theory, different from the usual homological ones present in the fixed point theory. An application to boundary value problems is provided as well as the brief exposition of basic facts from the cohomotopy theory. Mathematics Subject Classifications (2000) 47H04, 47A53, 55M20, 55Q55, 34A60.     

Boundary Solutions of Differential Inclusions and Recovering the Initial Data

The solutions of differential inclusions staying at the boundary of attained sets are investigated using the contingent cones to attained sets. The possibility of recovering the starting point and the time which elapsed since the beginning is shown when some fragments of an attained set are known.     

Existence of periodic solutions

An apparatus for proving existence theorems for periodic solutions of equations with discontinuous right-hand side and differential inclusions is developed. Translated fromMatematischeskie Zametki, Vol. 61, No. 5, pp. 769–784, May, 1997. Translated by V. N. Dubrovsky