This course is an introduction to the Monge-Ampere equation. After introducing basic concepts such as normal mapping and Monge-Ampere measure, we will study important properties of the Monge-Ampere measure including: weak continuity and invariance property. We then study the celebrated Aleksandrov's maximum principle, the comparison principle, John's lemma and applications. Finally, we will discuss the Dirichlet problem and sections of convex solutions to the Monge-Ampere equation.
The Monge-Ampere equation appears in many areas and applications including affine geometry, convex geometry, optimal transportation and meteorology. The Monge-Ampere equation and its applications is a very active area of research. This course hopes to provide solid background and motivate interested students entering this research area.
・Be familiar with basic concepts in the Monge-Ampere equation such as normal mapping, Monge-Ampere measure, Aleksandrov's solution
・Be familiar with modern tools and concepts in the Monge-Ampere theory such as John's lemma and sections of solutions
・Understand and be able to use maximum principles in the Monge-Ampere equation
normal mapping, Monge-Ampere measure, weak continuity, invariance property, Aleksandrov's solution, Aleksandrov's maximum principle, comparison principle, John's lemma, Dirichlet problem, sections of convex functions.
This is a standard lecture course. Homework will be assigned every week.
|第1回||The Monge-Ampere equation in different contexts; normal mapping, Monge-Ampere measure, Legendre transform||Details will be provided during each class|
|第2回||Aleksandrov's solution, Examples, Weak continuity of Monge-Ampere measure,||Details will be provided during each class|
|第3回||Invariances of the Monge-Ampere equation, maximum principles||Details will be provided during each class|
|第4回||Aleksandrov's maximum principle||Details will be provided during each class|
|第5回||John's lemma, comparison principle||Details will be provided during each class|
|第6回||The Dirichlet's problem: uniqueness and solvability by the Perron method||Details will be provided during each class|
|第7回||Sections of convex functions||Details will be provided during each class|
|第8回||Geometric properties of sections of solutions to the Monge-Ampere equation||Details will be provided during each class|
The course will be based on Part 3 of the instructor's lecture notes “N. Q. Le, The second boundary value problem of the prescribed affine mean curvature equation and related linearized Monge-Ampere equation”, available at: http://pages.iu.edu/~nqle/SBVP_lectures.pdf
Final exam 50%, assignments 50%.
Basic knowledge on advanced analysis is essential