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
CauchyKovalevskaya theorem:
Analytic functions, CauchyKovalevskaya theorem, change of variables, characteristic surface, wellposedness, 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) 3982510. 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 takehome midterm exam 20%, and the final project 40%.
Homework
Assigned and graded roughly every other week.
Midterm exam
The midterm will be a takehome 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.
