The boundary value problem of Poisson's equation is mainly discussed. The course begins with the explicit representation of a solution in some special cases, then introduces fundamental facts such as the mean value property and the maximum principle, and finally reach at the existence and the uniqueness of classical solutions.
This course is continuous with Advanced topics in Analysis B1.
Students are expected to understand the existence, uniqueness and fundamental properties of solutions of linear elliptic partial differential equations.
Laplace's equation, Poisson's equation, boundary value problem, Perron's method
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
Standard lecture course
Course schedule | Required learning | |
---|---|---|
Class 1 | Laplace's equation and Poisson's equation | Details will be provided during each class session. |
Class 2 | The Newtonian potential | Details will be provided during each class session. |
Class 3 | The Poisson kernal | Details will be provided during each class session. |
Class 4 | The mean value property and the maximum principle | Details will be provided during each class session. |
Class 5 | Harnack's inequality | Details will be provided during each class session. |
Class 6 | Perron's method 1 | Details will be provided during each class session. |
Class 7 | Perron's method 2 | Details will be provided during each class session. |
Class 8 | General second order linear elliptic equations | Details will be provided during each class session. |
None in particular
D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer-Verlag
Students need to submit a report. Details will be announced during the lecture.
Students are required to complete Advanced topics in Analysis B1 (MTH.C406).
None in particular