【作者】 姜亦成；

【导师】 付永强；

【作者基本信息】 哈尔滨工业大学 ， 基础数学， 2007， 硕士

【摘要】 在过去十年里,变指数函数空间理论及其应用引起了越来越多的研究者的兴趣.最初,变指数Lebesgue空间和Sobolev空间是被Kovacik和Rakosnik作为一种处理带有非标准增长和强制假设的非线性Dirichet边值问题的新方法引入的,这一课题有着重要的物理背景,被应用到电流流体学的研究中.变指数函数空间理论的研究得益于弹性力学、流体力学、具有p ( x )增长条件的变分法和微分方程等问题的研究.如今,具有变指数的变分问题及微分方程被国际上许多学者集中进行了研究,出现了许多成果.本文主要介绍变指数函数空间理论的最新成果,并提出一些悬而未决的问题,给出变指数函数空间理论在偏微分方程上的应用.本文第一章介绍此课题的研究背景,给出了变指数函数空间中的一些基本性质和结论,作为全文的预备知识.第二章主要研究了变指数空间中的Hardy-Littlewood极大算子的有界性、对偶空间以及内插问题,并给出了几个仍未解决的问题.第三章研究了变指数Sobolev空间中的紧性问题和嵌入问题.第四章在变指数函数空间框架下,给出了几类微分方程解的存在性条件.更多还原

【Abstract】 Last decade, more and more researchers have been interested in the theory of the variable exponent function space and its applications. In the beginning, Kovacik and Rakosnik introduced variable exponent Lebesgue and Sobolev spaces as a new method for dealing with nonlinear Dirichet boundary value problems with nonstandard growth and coercive assumption. The variable exponent spaces have an important physics background, and they are used in the study of electrorheological fluids. The study of these spaces has been stimulated by problems of elasticity, fluid dynamics, calculus of variations, and differential equations with p ( x )-growth conditions. Nowadays, variational problems and differential equations with variable exponent are intensively developed by many researchers worldwidely. A lot of results corresponding the researches have been attained. The paper introduces the recent results about the variable exponent function spaces, raises many open problems, and gives some applications on the partial differential equations.In this paper, chapter 1 introduces the background of the topic, and gives many properties and conclusions of the variable exponent function spaces as the preliminary knowledge. Chapter 2 does research on the boundedness of the Hardy-Littlewood maximal operators in the variable exponent Lebesgue spaces, its dual space, and the interpolation problems. Chapter 3 studies the problems of the compactness and the embedding in the variable exponent Sobolev spaces. Chapter 4 gives the conditions about the existence of the solutions of several differential equations in the variable exponent function space.更多还原