# Math 580

**Catalog description**

Classification and wellposedness of linear and nonlinear partial differential equations;
energy methods; Dirichlet principle.
Brief introduction to distributions; weak derivatives.
Fundamental solutions and Green's functions for Poisson equation, regularity, harmonic functions, maximum principle.
Representation formulae for solutions of heat and wave equations, Duhamel's principle.
Method of characteristics, scalar conservation laws, shocks.

**Topics to be covered**

*Cauchy-Kovalevskaya theorem*:
Analytic functions, Cauchy-Kovalevskaya theorem, change of variables, characteristic surface, well-posedness, classifications
*Quasilinear first order equations*:
Method of characteristics, local theory, conservation laws, inviscid Burgers' equation, viscosity solution, weak solution, shock waves, jump conditions, entropy criteria
*Harmonic functions*:
Green's identities, fundamental solution of the Laplace operator, uniqueness theorems, Green's function,
mean value property, Harnack inequalities, derivative estimates and analyticity,
maximum principles, Poisson's formula, removable singularity theorem,
Harnack convergence theorems
*Second order elliptic equations*:
Dirichlet problem, Dirichlet principle,
Poincaré's method of sweeping out, boundary regularity, barriers,
Schauder estimates, method of continuity, nonlinear examples
*Parabolic equations*:
Cauchy problem for the heat equation, Duhamel's principle, maximum principle, uniqueness, regularity, nonlinear equations
*Hyperbolic equations*:
Cauchy problem for the wave equation, Huygens principle, Duhamel's principle, energy method, symmetric hyperbolic systems

**Prerequisites**

MATH 375 or equivalent.

**References**

Lawrence Craig Evans, *Partial differential equations*. AMS 1998.
Fritz John, *Partial differential equations*. Springer 1982.
Jürgen Jost, *Partial differential equations*. Springer 2007.
Oliver Dimon Kellogg, *Foundations of potential theory*. Dover 2010.
François Trèves, *Basic linear partial differential equations*. Dover 2006.
David Gilbarg and Neil Sidney Trudinger, *Elliptic partial differential equations of second order*. Springer 2001.

**Lectures**

**TuTh 8:35am–9:55am**,
Burnside Hall 920

**Instructor**

Dr. Gantumur Tsogtgerel

Office: Burnside Hall 1123. Phone: (514) 398-2510. Email: gantumur -at- math.mcgill.ca.

Office hours: Just drop by or make an appointment

**Grading**

The final course grade will be the weighted average of homework 40%, the take-home midterm exam 20%, and the final project 40%.

**Homework**

Assigned and graded roughly every other week.

**Midterm exam**

The midterm will be a take-home exam.

**Final project**

The final project consists of the student reading a paper or monograph on an advanced topic, typing up notes, and presenting it in class as a short lecture.
A list of topics for the final project will be given out after the midterm exam.