The Basic Information of the Course
Course Number:202012420015
The Chinese Name of the Course:Sobolev空间
The English Name of the Course:Sobolev Spaces
In-class Hours and Allocation:total class hours:32, classroom teaching:23 class hours,classroom discussion:9 class hours
Credit(s):2
Semester:1
Applied Discipline:Mathematics
Course Object Oriented:Academic Master,Academic Doctor
Evaluation Mode: Curriculum design
Teaching Method:Combination of teaching and discussion at classroom as well as extracurricularresearch reports
Course Opening Department:College of Mathematical Sciences
Ⅱ. Prerequisite Course
Advanced mathematics or mathematical analysis, functional analysis
III. The Objectives and Requirements of the Course
The Objectives of the Course：Through the study of this course, students are required to master the basic contents and basic methods of Sobolev function space, and be able to apply these contents and methods in combination with the characteristics of partial differential equations.
The Requirements of the Course：This course requires students to use the basic knowledge of Sobolev space to analyze and deal with basic partial differential equation problems, read related literature and apply the related skills of Sobolev space to solve simple theoretical problems.
IV. The Content of the Course
Chapter 1 Relevant properties of/space
This chapter includes:/inequality and Minkowski’s inequality, completeness of space, separability of space and denseness of continuous functions in space,global continuity of functions in space, strong column compactness of sets,weak column compactness of bounded sets of modules in space.
Chapter 2 Mollifiers and Meanapproximation
This chapter includes:Properties of mollifiers, Mean approximation theorem, Unit decomposition theorem,about local space.
Chapter 3 Generalized derivatives and Sobolev space
This chapter includes: the general theory of the properties of generalized derivatives, the relation between the existence of generalized derivative and the absolute continuity of function, Sobolev function space and, function boundary in space.
Chapter 4 Embedding theorem
This chapter includes:some properties of Lipschitz type domain,basic approximation theoremand consistency of and ,some inequalitiesin , Sobolev embedding theorem,complete continuity of embedded operators.
Course for Ideological and Political Education
In terms of ideological and political education, we will carefully design and implement the teaching of each lesson, carry outthe work about "course ideology" in the entire process of education and teaching, and strive to achieve the organic unity of knowledge imparting, ability training and value guidance.The application of Sobolev space has a strong physical background and is closely connected with the real world. For example, we can use the technique of parameter assignment in Sobolev space to conduct in-depth research on several types of equations from the field of physics.Specifically, the transport equations from radiation fluid mechanics have studied the asymptotic limit problem of mixed layer equations by using the asymptotic expansion method and the energy method combined with the parameter Sobolev space technique.Therefore, the in-depth analysis of Sobolev space and its applications can enable students to understand the physical meaning corresponding to the models in mathematics, which thereby inspires students' love for scientific research, establishes correct values, and engages in scientific research as soon as possible.
The following will provide two specific examples.
The first example.The mathematician Wu Xinmou is one of the main founders of the research on partial differential equations in China. He organized the differential equations laboratory of the Institute of Mathematics, Chinese Academy of Sciences, compiled the first textbook of partial differential equation theory in China, and presided over a large-scale national workshop. A large number of top talents in the field of partial differential equations such as Ding Xiaqi, Wang Guangyin, Sun Hesheng, Qiu Peizhang have been trained, and they have made significant contributions to the construction of the research team of partial differential equations in China.
Professor Wu Xinmou studied abroad in France in his early years and was engaged in the research of fluid mechanics, focusing on the stability study of viscous fluid motion. The results he obtained are of great significance for overcoming the difficulties faced by the small motion method commonly used in classical fluid motion stability theory. Since the 1950s, Professor Wu Xinmou has focused on the research of mixed equations and created a "zero integral" method, whichwas used by the Courant Institute of the United States as a designated reading document for graduate students.
After the founding of the people's Republic of China in 1949, Professor WuXinmou overcame many difficulties and returned to China with his family in 1951. He devoted himself enthusiastically to the cause of mathematics in the people's Republic of China. Under almost blank conditions, a seminar on PDE was set up, which is the first large-scale seminar in China with the theme of modern PDE theory. About 100 teachers were sent to the seminar by colleges and universities across the country, including Gu Chaohao, Qi Minyou, Xiao Shutie, Wu Zhuoqun, Dong Guangchang, etc. Thelecture notesprepared by Wu xinmou and used in the workshop was officially published soon. It is the first specialized textbook of partial differential equation theory after the establishment of the people's Republic of China, and also a distinctive research reference book. Since then, many mathematics departments of colleges and universities have offered the course of partial differential equation theory, and the backbone team of partial differential equation work nationwide has begun to form.
The fruitful results in this course are inseparable from the contributions of many contemporary Chinese mathematicians. Therefore, through the introduction of early Chinese mathematicians, students will understand the history and current situation of the development of this course, learn from the spirit of hard work of the older mathematicians in the turbulent international situation, increase the cultural confidence and pride of young students,andprovide spiritual motivation for efforts to study scientific and cultural knowledge.
The second example.Professor Ding Xiaqi, academician of the Chinese Academy of Sciences, is a well-known mathematician in mathematics at home and abroad. He is the first generation of mathematics workers trained and developed by New Chinaitself, and alsoisone of the most accomplished mathematicians in the field of partial differential equations inChina. He has made many important research achievements and made outstanding contributions to the development of my country's mathematics.
Professor Ding Xiaqi has been working in the Institute of Mathematics of the Chinese Academy of Sciences since graduating from university. For decades, he insisted on working late into the night every day. Even during hospitalization due to illness, he still insisted on academic research and never stopped. In the decades of work, Professor Ding Xiaqi has long been devoted to the research of partial differential equations, function space, number theory, mathematical statistics, harmonic analysis and numerical analysis. He is most successful in the space of partial differential equations and functions. His work on mixed equations, elliptic systems, and discontinuous solutions has been highly praised by colleagues at home and abroad. In particular, he and his collaborators solved the well-known mathematical problem-the study of the global solution of isentropic airflow. Completed the "Compensation Column Tightening Principle and Isentropic Gas Dynamics Equations" project, which caused a strong response in the international mathematics community and was highly evaluated by many famous mathematicians including P.D. Lax and J. Glimm. In addition, for decades, Ding Xiaqi has always attached great importance to the cultivation of talents. Some of the students he trained and comrades who worked under his influence or help for a long time have become doctoral tutors in various domestic universities, and some have become internationally renowned mathematicians, such as Professor Chen Guiqiang.
The research results of parabolic equation and embedding theorem involved in this course also agglomerate the wisdom of a group of Chinese mathematicians. Therefore, through the introduction of the contemporary Chinese mathematicians, the students can understand the research results of the Chinese mathematicians related to this course, learn the firm belief that the older generation mathematicians are still engaged in mathematics research under the tough living environment, learn the perseverance and never give up spirit of the older generation mathematicians, and encourage the young students to set up lofty aspirations and study hard.
V. Reference Books, Reference Literatures, and Reference Materials
A. Text Books, Monographs and References
1.刘亚成,徐润章.Sobolev空间及其在偏微分方程中的应用.黑龙江科技出版社,2005年.
2.王明新.索伯列夫空间.高等教育出版社,2013年.
3.Lawrence C.Evans. Partial Differential Equations. American Mathematical Society,2003.
4.Haim,Brezis. Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, 2015.
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