The main subject of this course is to understand power concavity properties of solutions to elliptic and parabolic equations. After introducing the notion of power concavity, we study the concavity maximum principle to obtain power concavity properties of solutions to elliptic and parabolic equations. Furthermore, we study non-preservation of quasi-concavity by the Dirichlet heat flow, preservation of log-concavity for parabolic equations, parabolic power concavity and their related topics.
Power concavity is a useful notion to describe the shape of solutions to partial differential equations, in particular, elliptic and parabolic equations. Thanks to the concavity maximum principle, power concavity properties for elliptic and parabolic equations have been studied by many mathematicians about for 40 years. Recently, power concavity properties were developed extensively via viscosity solutions. The aim of this lecture is to understand the outline of the recent development of power concavity properties of solutions to elliptic and parabolic equations.
・Understand the definition of power concavity and its properties
・Understand the concavity maximum principle
・Understand the outline of recent development of power concavity properties of solutions to elliptic and parabolic equations
power concavity, concavity maximum principle, log-concavity, parabolic power concavity, Minkowski addition
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. There will be some assignments.
|Course schedule||Required learning|
|Class 1||The following topics will be covered in this order : - power concavity and its properties - concavity maximum principle and its applications - non-preservation of quasi-concavity - preservation of log-concavity for parabolic equations - parabolic power concavity - parabolic equations and Minkowski addition - analysis of power concavity on manifolds||Details will be provided during each class session.|