# Spring 2018

## Instructor Peter Kuchment

Office Rm. Blocker 614A

E-mail: kuchment@math.tamu.edu

Section: 600, Time: TR 12:45-2:00pm, Room: BLOC 605AX
Textbook: L. C. Evans, Partial Differential Equations: 2nd edition, American Math. Society, 2010.
Office hours: TR 2:30 - 3:30, Blocker 614A
Additional office hours can be arranged by appointment.

## Introduction

This class is devoted to the theory of linear partial differential equations. Assuming initial familiarity of students with classical PDEs (Laplace, wave, and heat equation), the following main topics will be addressed:

1. Sobolev spaces

2. Classification of equations of 2nd order

3. Second order elliptic equations (including boundary value problems theory and spectral theory)

4. Linear evolution equations (including initial-and-boundary-value parabolic and hyperbolic problems)

Besides numerous applications inside mathematics, the PDEs form the core part of our scientific understanding of the physical world: from physics to chemistry, to biology, to meteorology, you name it.

The class will be based on the second part "Theory for linear partial differential equations" of the well respected recent textbook by L. Evans.

## Prerequisite:

MATH 611, and Real Analysis, or their equivalents, or instructor's consent (if unsure, please contact the instructor).

## Assignments

Grading will be based on home assignments and a take-home final exam.

## Tentative schedule of the course(watch for updates)

Week

Topics and sections

Home assignments

Exams

01/16 - 01/30

Chapter 5: Sobolev spaces

TBA

n/a

01/30-02/06

Classification of 2nd order PDEs.

TBA

n/a

02/06-03/08

Chapter 6: Elliptic problems

TBA

n/a

03/13- 03/15

Spring break

Have some rest :-)

n/a

03/20-04/26

Chapter 7: Evolution equations

na

Take home final, due by May 7th

Percentage of points

90% and higher

A

80% and higher

B

70% and higher

C

60% and higher

D

Less than 60%

F

• L. Bers, F. John, and M. Schechter, Partial Differential Equations, Interscience, New York, 1964. American Mathematical Society, 1979. A wonderful (albeit old) book introducing properties of main types of PDEs, including Sobolev spaces.

• Richard Courant and David Hilbert, Methods of Mathematical Physics, two volume set (any edition). This is an absolutely wonderful classics. In spite of being half of a century old, it is still a very important book, in many instances not surpassed by anything else. A must reading for anyone using PDEs extensively.

• Fritz John, Partial Differential Equations, Springer Verlag. Although outdated and more limited that Evans' book, this is still a very good introduction to PDEs (mostly Math611).

• Andras Vasy, Partial Differential Equations: An Accessible Route through Theory and Applications, AMS 2015. A good new book, with a different approach.

• R. Strichartz, A guide to distribution theory and Fourier transform, WorldSci, several editions. A wonderful introduction to distribution theory and Fourier transform.

• Michael E. Taylor, Partial Differential Equations: Basic Theory (Texts in Applied Mathematics, 23), Springer Verlag, and its two consecutive volumes. A comprehensive set of books covering all major topics of PDEs from contemporary points of view. Very geometric, in most cases equations are considered on manifolds.

• Gregory Eskin, Lectures on Linear Partial Differential Equations, AMS 2011. Another good contemporary textbook written by a well known expert, the approach is different.

• David Gilbarg and Neil S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag (latest edition is 2001). A classic tretease on second order elliptic equations.

• W. D. Evans and D. E. Edmunds, Spectral Theory and Differential Operators (Oxford Mathematical Monographs), Oxford Univ Press 1997. A terrific advanced book on Sobolev spaces, elliptic differential operators, and spectral theory.

• L. Hormander, The Analysis of Linear Partial Differential Operators, Springer Verlag. A 4 volume set of fundamental books written by the world leading expert and covering most important techniques and results on linear PDEs (for an advanced reader).

Make-ups: for missed quizzes, home assignments and exams will only be allowed for a university approved excuse in writing. Wherever possible, students should inform the instructor before an exam or quiz is missed. Consistent with the University Student Rules: students are required to notify an instructor by the end of the next working day after missing an exam or quiz. Otherwise, they forfeit their rights to a make-up.
Grade complaints: Sometimes the instructor might make a mistake grading your work. If you feel that this has happened, you have one week since the graded work was handed back to you to talk to the instructor. If a mistake is confirmed, the grade will be changed. No complaints after that deadline will b e considered.
Copying work done by others, either in-class or out of class, is an act of scholastic dishonesty and will be prosecuted to the full extent allowed by University policy. Collaboration on assignments, either in-class or out-of-class, is forbidden unless permission to do so is granted by your instructor. For more information on university policies regarding scholastic dishonesty, see University Student Rules.
All printed materials disseminated in class or on the web are protected by Copyright laws. One xerox copy (or download from the web) is allowed for personal use. Multiple copies or sale of any of these materials is strictly prohibited.