【课程】 2015年7月6日至7月31日（July 6-July 31, 2015）
【课程名称】(COURSE TITLE) 偏微分方程的自由边值问题 (Free Boundary Problems in PDE)
【授课教师】(COURSE Instructor) 姓名：胡钡 (Bei Hu) 来自学校：圣母大学应用数学计算统计系教授
【授课对象】(AUDIENCE) 本科大四生、研究生、博士后、青年教师
【授课语言】(LANGUAGE OF INSTRUCTION) 中英双语 Chinese and English
【先修课程】(PREREQUISITES) (Undergraduate) Functional Analysis, Complex analysis, PDE.
【授课安排】 (Course Arrangement)

授课时间：7月6日--7月31日，每周授课8小时： 星期一、二、四、五，上午9：30到11：30。 其它时间另有安排，包括部分来访学者的系列学术演讲。 Time arrangement: The course will start on Monday July 6th and end on Friday July 31st. The class will meet for 8 hours per week: Two hours (9:30-11:30am) each on Monday, Tuesday, Thursday and Friday. Other activities will be arranged, including lecture series by various visiting scholars.

【联系方式】 (Contact Information)

主办单位：中国人民大学数学科学研究院 （Institute for Mathematical Sciences, RUC）
主办方联系人: Yuan Lou, lou.8@osu.edu; Jun Wang, wangjunruc@ruc.edu.cn

【教师简介】(ABOUT THE COURSE INSTRUCTOR)

Prof. Dr. Bei Hu graduated with a Bachelor at East China Normal University. He received his Ph.D. from the University of Minnesota in 1990. He worked at University of Notre Dame and has been a Full professor since 1999. He is one of the leading experts in studying various applications of PDE theory. He made significant contributions in Blowup theory, Mathematical Finance, and Mathematical Biology. He published an entry level lecture notes for graduate students: Bei Hu, Blow-up Theories for Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 2018 (Book series), Springer, New York, New York, 2011. Dr. Hu has published over 80 papers and his papers (as of Nov 2014) were cited more than 1300 times. He had over 20 years of teaching experience and received teaching awards at the University of Notre Dame twice.

【课程简介】(COURSE DESCRIPTION)

One common feature for problems in Mathematical Finance and Mathematical Biology is that they involve a boundary that is not known in advance, that is so-called a free boundary. The boundary of a tumor, for example, is changing in time because the tumor may grow or shrink. Such a problem does not fall into the standard category of boundary value problems of partial differential equation (PDE), while such a problem represents a significant application of mathematical theory. This is actually the beauty of PDE (and Mathematics in general)!

This course serves as an introduction to free boundary problems arising from various applications. Mathematical theory for free boundary problems will be presented, various examples will be studied. The examples include the famous classical Stefan problem which models any material with a liquid phase and a solid phase (such as water and ice). As an interesting example, the growth of a tumour will be modelled as a free boundary problem and studied through mathematical analysis. The course will build the mathematical theory required for such studies.

【课程大纲】(COURSE SYLLABUS)
  1. Linear elliptic PDE theory.

• Maximum principles for elliptic equations;
• Linear PDE;
• Barrier function;
• Holder space;
• Schauder estimates and continuation method for second order elliptic PDEs;
• Solvability of the Dirichlet problem;

 2. Linear parabolic PDE theory.

• Maximum principles for parabolic equations;
• Linear PDE;
• Holder space;
• Schauder estimates and continuation method for second order elliptic PDEs;
• Solvability of the Dirichilet problem;
• Asymptotic behaviour;
• Liyapunov function;

 3. Calculus of variation.

• Weak derivatives;
• Dirichlet principle;
• Variational inequality and Obstacle problem – a free boundary problem;
• One phase Stephan problem as an obstacle problem;

 4. Fixed point theorems.

• Contraction mapping principle;
• Schauder fixed point theorem;
• Leray-Schauder fixed point theorem;
• Applications of fixed point theorem to PDE;

 5. A simple cell growth model.

• The model;
• Existence;
• The free boundary;
• Stability;

 6. A tumor growth model;

• The model;
• Radially symmetric solutions;
• The free boundary;
• Stability;

【参考读物】 (References)
 1. Ya-Zhe Chen, and Lan-Cheng Wu, 1998, Second Order Elliptic Equations and Elliptic Systems, AMS Translation of Mathematical Monographs, Vol 174, AMS.
 2. 陈亚浙：二阶抛物型偏微分方程。北京大学出版社，2003年.
 3. Avner Friedman， Variational Principles and Free Boundary Problems. Dover, 2010.
 4. Bei Hu: Lecture notes will be provided