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2018　Special courses on advanced topics in Mathematics E

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Academic unit or major
Mathematics
Instructor(s)
Tonegawa Yoshihiro  Kohsaka Yoshihito 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive ()  
Group
-
Course number
ZUA.E335
Credits
2
Academic year
2018
Offered quarter
1Q
Syllabus updated
2018/3/20
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

The main theme of this lecture is analysis of geometrical evolution equations based on analytical method of nonlinear partial differential equations. As a geometrical evolution equation, let us consider the mean curvature flow as the object of analysis and consider the case where three hypersurfaces that move due to mean curvature flow intersect each other. First, we learn how to express mean curvature, and derive initial value and boundary value problem of system of nonlinear parabolic partial differential equations from the problem of mean curvature flow in the above setting. For the initial value and boundary value problem of the nonlinear parabolic partial differential equation obtained, applying the analytical method of nonlinear partial differential equation, we clarify the existence of time local solution.
In this lecture, we will learn how to derive geometric quantities for hypersurfaces and prove the existence of temporal local solutions of nonlinear parabolic partial differential equations.

Student learning outcomes

· The mean curvature can be derived for a given hypersurface.
· It is possible to understand linearization of nonlinear problem and the analytical method on the linearization problem.
· You can understand the existence proof of the time local solution of the nonlinear problem using the fixed point theorem.

Keywords

Hypersurface, mean curvature, parabolic partial differential equation, fixed point theorem

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

In an ordinary lecture format. Also, issue reports as appropriate.

Course schedule/Required learning

  Course schedule Required learning
Class 1 Mean curvature flow and derivation of mean curvature for hypersurface I. Details will be provided during each class session
Class 2 Mean curvature flow and derivation of mean curvature for hypersurface II.
Class 3 Mean curvature flow and derivation of mean curvature for hypersurface III.
Class 4 Mean curvature flow and derivation of mean curvature for hypersurface IV.
Class 5 Derivation of initial value and boundary value problem of nonlinear parabolic partial differential equation and its linearization I.
Class 6 Derivation of initial value and boundary value problem of nonlinear parabolic partial differential equation and its linearization II.
Class 7 Derivation of initial value and boundary value problem of nonlinear parabolic partial differential equation and its linearization III.
Class 8 Derivation of initial value and boundary value problem of nonlinear parabolic partial differential equation and its linearization IV.
Class 9 Analysis on the linearization problem I.
Class 10 Analysis on the linearization problem II.
Class 11 Analysis on the linearization problem III.
Class 12 Proof of existence of time local solution of nonlinear problem using fixed point theorem I.
Class 13 Proof of existence of time local solution of nonlinear problem using fixed point theorem II.
Class 14 Proof of existence of time local solution of nonlinear problem using fixed point theorem III.
Class 15 Proof of existence of time local solution of nonlinear problem using fixed point theorem IV.

Textbook(s)

Not specified.

Reference books, course materials, etc.

「Partial Differential Equations」 L. C. Evans, AMS
「Parabolic Boundary Value Problems」 S. D. Eidelman, N. Z. Zhitarashu, Birkhauser

Assessment criteria and methods

By evaluating reports.

Related courses

  • MTH.C341 ： Differential Equations I
  • MTH.C342 ： Differential Equations II

Prerequisites (i.e., required knowledge, skills, courses, etc.)

Students are expected to have passed Differential Equations I and Differential Equations II.

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