Differential Equations Examination Syllabus

Topics

The Laplace equation: Fundamental solution; Mean value formulas; Properties of harmonic functions; Green's function; Energy methods.

The heat equation: Fundamental solution; Mean value formula; Properties of solutions (Maximum principle); Energy methods.

The wave equation: Solution by spherical means; The non-homogeneous problem; Energy methods.

First-order PDEs: Characteristics (The linear, quasi-linear, and nonlinear cases); Hamilton-Jacobi equations.

Sobolev spaces (Introduction).

Second-order elliptic equations: Weak solutions (Existence and regularity); Maximum principles.

Selected Bibliography

1. L. C. Evans, Partial Differential Equations
2. P. Garabedian, Partial Differential Equations
3. F. John, Partial Differential Equations (4th Ed.)
4. Nicolae H. Pavel, Introduction to Partial Differential Equations - a graduate textbook.