$Department of Mathematics$

Graduate course MAT 1062

Topics in Linear Partial Differential Equations II

Professor Victor Ivrii

Monday	12:10 - 1:00	BA 2185
Wednesday	1:10 - 2:00	BA 4010
Friday	11:10 - 12:00	BA 6180

My office: Earth Sciences, Room 4141	My time-Table	Course Description
My phone: 416-978-4031	Announcements	MAT1062 Bulletin Board ^New
My email: ivrii@math.toronto.edu	Handouts	Computer Help Bulletin Board

Description

This course covers more advanced topics in PDE than those covered by MAT 1060 as given in 2004-2005. Students are assumed to have attended this course or undergraduate courses APM 346H or MAT 351Y and familiar with methods of solving of basic PDEs.

Content

Theory of Distributions.
- Classes D, E, S and their dual D', E', S'. Basic operations, Fourier transform.
- Sobolev spaces H^s on R^d.
- Paley-Wiener theorem.
Hyperbolic linear equations with constant coefficients
- Solution of the Cauchy problem for constant coefficient operators via Fourier transform.
- Hyperbolicity, Gårding hyperbolicity, strong hyperbolicity.
- Fundamental solution of the Cauchy problem. Support of solutions. Characteristic cone.
- Singular support of solutions: strongly hyperbolic case. Examples (wave equation, Maxwell, elasticity)
- Singular support of solutions: non-strongly hyperbolic case, conical refraction. Examples (crystall optics, elasticity, magnetic hydrodynamics).
Introduction to pseudo-differential operators.
- Pseudo-differential operators.
- Calculus.
- L²-boundedness.
- Application of pseudo-differential operators to elliptic equations. Parametrices.
Strictly hyperbolic equations and systems.
- Energy estimates. Existence theorem for Cauchy problem.
- Holmgren uniqueness theorem.
- Propagation of singularities. Examples (partially the same as before but in inhomogeneous media).
- Construction of oscillatory solutions.
Strictly hyperbolic equations and systems near boundary.
- Energy estimates, existence and uniqueness of solutions.
- Construction of oscillatory solutions.
- Reflection and refraction. Compensation waves.
- Gliding and grazing waves.
- Rayleigh waves.
- Crawling waves.

I will not follow any specific book but rather picking here and there, including such old books as R.Courant and D.Hilbert, Methods of mathematical physics, Vol I, II Interscience Publishers (this book is anything but obsolete).

I will try to make pretty complete handouts and launch bulletin board to discuss the course.

What you are supposed to know in advance

This is not the course for beginners and you need to have a knowledge of the material covered in one of the following PDE courses: APM346H, APM351Y, MAT1060. Knowledge of Real Analysis (MAT337 or better), Complex Analysis (MAT334 or better) and Ordinary Differential Equations (MAT244 or better) is assumed.

Evaluation

N presentation(s) (1≤N≤3, ∼45 min each) and
M report(s) covering some topic (in depth and with extras) with 0≤M≤3-N, ∼15 pp each; and they should be printed (electronic version [tex, pdf, dvi, ps] is very welcome!) or well hand-written and
L=3-N-M hours written exam (with the corresponding number of questions/problems).

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