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