双重非线性发展型方程及H-半变分不等式问题研究
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摘要
本文首先介绍非线性问题研究中一类重要的单调型算子—伪单调算子,讨论此类算子在非线性问题研究过程中的各种多值推广形式以及它们之间的相互关联.然后,结合偏微分方程、非线性单调算子、凸分析、非光滑分析、集值分析理论以及隐时间离散方法、Lebesgue-Bochner函数空间中的紧致性引理等,重点研究几类含有伪单调算子的双重非线性发展型方程及H-半变分不等式问题.首次建立了这些问题的解的存在性定理,推广了双重非线性发展型方程,以及抛物型、双曲型发展方程和H-半变分不等式问题的现有成果.
第一章为绪论部分,先引入不等式问题的两大分支—变分不等式和H-半变分不等式,简要介绍非线性问题的研究内容、难点和方法.随后,总结了非线性发展型方程以及椭圆型、抛物型和双曲型H-半变分不等式问题的已有成果和主要研究方法.最后介绍本文具体研究内容和主要理论成果.
第二章归纳了本文所需的基础知识和理论.主要包括函数空间、凸分析、非光滑分析、非线性单调算子理论、几个重要的紧性引理以及非线性泛函分析中的一些重要结论.
第三章第一部分重点介绍几类推广形式的多值伪单调映射和与之相关的抛物型问题的存在性定理.特别地,本节指出由Kasyanov等教授提出的一类Wλ0型伪单调映射和已有的广义伪单调映射等价而并非其推广.第二部分研究一类具有伪单调单值算子的非线性发展型方程.结合对偶算子正则化方法,在较弱的强制性条件下建立了其解的存在性定理.
鉴于有关双重非线性发展型方程以及椭圆、抛物和双曲型的H-半变分不等式问题理论已比较成熟,成果非常丰富,但有关双重非线性发展型方程的研究局限于两个非线性算子都是极大单调算子或凸函数的次微分形式,而发展型的H-半变分不等式问题的理论研究又只是针对抛物和双曲型问题,有关双重非线性型发展型的H-半变分不等式问题没有任何理论研究和成果.因此,本文从这一切入点着手,在随后四至六章中重点研究几类双重非线性发展型方程及H-半变分不等式问题.
在第四和第五章中,我们分别考虑了一类双重单值非线性发展型H-半变分不等式问题和一类双重非线性发展型边界H-半变分不等式问题.而第六章则主要研究一阶和二阶双重非线性发展型包含及其在H-半变分不等式问题中的应用,这章中非线性算子可以是多值型算子,其理论成果也更具普遍性.
有别于研究抛物型H-变分不等式的诸如Faedo-Galerkin方法、对偶算子的正则化法以及上下解方法等,本文采用隐时间离散技术将发展型问题椭圆化,借助椭圆型问题的存在性定理构造逼近解,通过一系列的先验估计和建立恰当的紧性引理,并结合Clarke广义次梯度以及单调型算子特性实现了逼近解到真解的极限过程.我们的研究推广了已有的理论成果,是这方面最新突破和发展.
In this thesis, we first introduce a class of monotone operators, namely, pseudomonotone operator, which plays an important role in the study of nonlinear problems. We present its various generalization forms and discuss their relations to each other. Then, several types of doubly nonlinear evolution equations and hemivariational inequali-ties with pseudomonotone operators are studied seriously by means of the theory of partial differential equations, nonlinear monotone oper-ators, convex analysis, nonsmooth analysis and multivalued analysis, the method of implicit time-discretization and the compact embed-ding theorems in Lebesgue-Bochner spaces. The existence theorems are established at the first time and our results generalize the exist-ing ones concerning doubly nonlinear evolution equations, parabolic, hyperbolic evolutional equations and hemivariational inequalities.
Chapter One is a general introduction. It introduces the two branches of inequalities, i.e., variational inequalities and hemivari-ational inequalities, and presents the main difficulty, methods and techniques regarding nonlinear problems. Then, we recall the main methods and results concerning nonlinear evolution equations, ellip-tic, parabolic and hyperbolic equations and hemivariational inequal-ities. Finally, we show the main point of this paper and the work of the following chapters.
Chapter Two is concerned with the preliminaries necessary for the study of this paper, including the theory of functional spaces, convex analysis, nonsmooth analysis and monotone operators, sev-eral compactness lemmas and some conclusions in nonlinear functional analysis.
In the first section of Chapter Three, we introduce some gener- alized definitions of multivalued pseudomonotone mapping and the existence theorems for parabolic problems governed by them. Par-ticularly, we show that the Wλ0pseudomonotone mapping given by professor Kasyanov is not the generalization of its previous one. The second section deals with a class of nonlinear evolution equations with pseudomonotone operators. The existence theorems of solutions for these problems are established under a weaker coerciveness condition compared to the previous result.
As we know, much concern is devoted to doubly nonlinear evolu-tion equations and elliptic, parabolic, hyperbolic hemivariational in-equalities and the theory result is fruitful. However, on the one hand, the study of doubly nonlinear equations requires that both the two nonlinear operators be maximal monotone and the subdifferential of convex functional. On the other hand, the evolution hemivariational inequalities studied by mathematicians now are mainly parabolic and hyperbolic. Therefore, considering these situations, in this thesis we deal with several classes of doubly nonlinear evolution equations and hemivariational inequalities in the following fourth to sixth chapter.
Chapter Four and Five are concerned with a class of single-valued doubly nonlinear evolutional hemivariational inequalities and a class of doubly nonlinear evolutional hemivariational inequalities of bound-ary type, respectively. In Chapter Six, we deal with the doubly non-linear evolutional inclusions and hemivariational inequalities of first and second orders. In this chapter, the nonlinear operator may be multivalued and the results are more general.
Unlike the methods to parabolic hemivariational inequalities such as the method of Faedo-Galerkin, regularity method of duality opera-tor, and upper and lower solution method, in this paper we adopt the implicit time-discretization technique to transform the evolution prob-lems into elliptic ones, and then construct the approximate solutions by the existence theorems of elliptic problems. After establishing a series of a priori estimates and appropriate compact lemmas, we com-plete the limit process of approximate solutions to true solutions by using the properties of Clarke's generalized gradient and the mono-tonicity of the operators. Our results extend the existing ones and represent the latest breakthrough in this field.
第一章为绪论部分,先引入不等式问题的两大分支—变分不等式和H-半变分不等式,简要介绍非线性问题的研究内容、难点和方法.随后,总结了非线性发展型方程以及椭圆型、抛物型和双曲型H-半变分不等式问题的已有成果和主要研究方法.最后介绍本文具体研究内容和主要理论成果.
第二章归纳了本文所需的基础知识和理论.主要包括函数空间、凸分析、非光滑分析、非线性单调算子理论、几个重要的紧性引理以及非线性泛函分析中的一些重要结论.
第三章第一部分重点介绍几类推广形式的多值伪单调映射和与之相关的抛物型问题的存在性定理.特别地,本节指出由Kasyanov等教授提出的一类Wλ0型伪单调映射和已有的广义伪单调映射等价而并非其推广.第二部分研究一类具有伪单调单值算子的非线性发展型方程.结合对偶算子正则化方法,在较弱的强制性条件下建立了其解的存在性定理.
鉴于有关双重非线性发展型方程以及椭圆、抛物和双曲型的H-半变分不等式问题理论已比较成熟,成果非常丰富,但有关双重非线性发展型方程的研究局限于两个非线性算子都是极大单调算子或凸函数的次微分形式,而发展型的H-半变分不等式问题的理论研究又只是针对抛物和双曲型问题,有关双重非线性型发展型的H-半变分不等式问题没有任何理论研究和成果.因此,本文从这一切入点着手,在随后四至六章中重点研究几类双重非线性发展型方程及H-半变分不等式问题.
在第四和第五章中,我们分别考虑了一类双重单值非线性发展型H-半变分不等式问题和一类双重非线性发展型边界H-半变分不等式问题.而第六章则主要研究一阶和二阶双重非线性发展型包含及其在H-半变分不等式问题中的应用,这章中非线性算子可以是多值型算子,其理论成果也更具普遍性.
有别于研究抛物型H-变分不等式的诸如Faedo-Galerkin方法、对偶算子的正则化法以及上下解方法等,本文采用隐时间离散技术将发展型问题椭圆化,借助椭圆型问题的存在性定理构造逼近解,通过一系列的先验估计和建立恰当的紧性引理,并结合Clarke广义次梯度以及单调型算子特性实现了逼近解到真解的极限过程.我们的研究推广了已有的理论成果,是这方面最新突破和发展.
In this thesis, we first introduce a class of monotone operators, namely, pseudomonotone operator, which plays an important role in the study of nonlinear problems. We present its various generalization forms and discuss their relations to each other. Then, several types of doubly nonlinear evolution equations and hemivariational inequali-ties with pseudomonotone operators are studied seriously by means of the theory of partial differential equations, nonlinear monotone oper-ators, convex analysis, nonsmooth analysis and multivalued analysis, the method of implicit time-discretization and the compact embed-ding theorems in Lebesgue-Bochner spaces. The existence theorems are established at the first time and our results generalize the exist-ing ones concerning doubly nonlinear evolution equations, parabolic, hyperbolic evolutional equations and hemivariational inequalities.
Chapter One is a general introduction. It introduces the two branches of inequalities, i.e., variational inequalities and hemivari-ational inequalities, and presents the main difficulty, methods and techniques regarding nonlinear problems. Then, we recall the main methods and results concerning nonlinear evolution equations, ellip-tic, parabolic and hyperbolic equations and hemivariational inequal-ities. Finally, we show the main point of this paper and the work of the following chapters.
Chapter Two is concerned with the preliminaries necessary for the study of this paper, including the theory of functional spaces, convex analysis, nonsmooth analysis and monotone operators, sev-eral compactness lemmas and some conclusions in nonlinear functional analysis.
In the first section of Chapter Three, we introduce some gener- alized definitions of multivalued pseudomonotone mapping and the existence theorems for parabolic problems governed by them. Par-ticularly, we show that the Wλ0pseudomonotone mapping given by professor Kasyanov is not the generalization of its previous one. The second section deals with a class of nonlinear evolution equations with pseudomonotone operators. The existence theorems of solutions for these problems are established under a weaker coerciveness condition compared to the previous result.
As we know, much concern is devoted to doubly nonlinear evolu-tion equations and elliptic, parabolic, hyperbolic hemivariational in-equalities and the theory result is fruitful. However, on the one hand, the study of doubly nonlinear equations requires that both the two nonlinear operators be maximal monotone and the subdifferential of convex functional. On the other hand, the evolution hemivariational inequalities studied by mathematicians now are mainly parabolic and hyperbolic. Therefore, considering these situations, in this thesis we deal with several classes of doubly nonlinear evolution equations and hemivariational inequalities in the following fourth to sixth chapter.
Chapter Four and Five are concerned with a class of single-valued doubly nonlinear evolutional hemivariational inequalities and a class of doubly nonlinear evolutional hemivariational inequalities of bound-ary type, respectively. In Chapter Six, we deal with the doubly non-linear evolutional inclusions and hemivariational inequalities of first and second orders. In this chapter, the nonlinear operator may be multivalued and the results are more general.
Unlike the methods to parabolic hemivariational inequalities such as the method of Faedo-Galerkin, regularity method of duality opera-tor, and upper and lower solution method, in this paper we adopt the implicit time-discretization technique to transform the evolution prob-lems into elliptic ones, and then construct the approximate solutions by the existence theorems of elliptic problems. After establishing a series of a priori estimates and appropriate compact lemmas, we com-plete the limit process of approximate solutions to true solutions by using the properties of Clarke's generalized gradient and the mono-tonicity of the operators. Our results extend the existing ones and represent the latest breakthrough in this field.
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