To thoroughly implement the goal of cultivating top-tier innovative talents, stimulate undergraduates' research potential, and enhance their academic literacy and practical research capabilities, the "Scaling Plan" Analysis and Equations Seminar has been conducted in an orderly and sustained manner. Centered on authoritative textbooks such as Evans' Partial Differential Equations, supplemented by presenters' insights, the seminar series covers foundational theories including partial differential equations and functional analysis, aiming to provide undergraduates with a high-level platform for academic exchange and research training.
Held every Sunday morning, the seminar adopts a student-led format with guidance from senior faculty, fostering a dynamic teacher-student interaction.
First Session
Led by Fang Zijin (2023, Mathematics and Applied Mathematics, Self-Strengthening Class) under Professor Zhou Yifu's supervision, the session focused on the mean value property of Laplace equations, deriving average inequalities for subharmonic functions on spheres and spherical surfaces, laying the groundwork for subsequent Perron method discussions. Dean Fan Huijun of the School of Mathematics and Statistics provided guidance, illustrating with examples that subharmonic functions in the simplest case are convex functions, making their average inequalities intuitively clear. Fang also employed mollification techniques to prove the smoothness of harmonic functions and derived the solution to the Dirichlet boundary value problem for Laplace equations in spherical domains using integration by parts and Green's function methods, expanding students' problem-solving perspectives.
Second Session
Fang continued as the presenter, with Professor Wang Kelei guiding the discussion on the Perron method. Fang systematically explained the maximum principle and Harnack inequality, further revealing the compactness of harmonic functions and attempting a rigorous proof of the Perron method. Despite encountering challenges, Dean Fan's meticulous guidance helped advance the proof, which ultimately applied the barrier function method to rigorously establish the existence and uniqueness of solutions to the Dirichlet problem under ideal boundary conditions. Professor Wang also assigned targeted exercises. Participants recognized that gradient estimates and Harnack inequalities together imply the compactness of harmonic functions, while the Harnack inequality for heat equations is typically derived from gradient estimates, showcasing deep theoretical connections within analysis.
Third Session
He Tinghao (2024, Mathematics) presented under Professor Yang Chenglang's supervision, focusing on the solution to the Cauchy problem for heat equations. Using Fourier transforms as the core tool, He discussed solutions for both homogeneous and inhomogeneous cases, verifying regularity and boundary conditions through step-by-step back-substitution. Professor Yang emphasized the importance of interchanging variable derivatives and integration by parts in linear PDEs, significantly broadening students' understanding of solution regularity.
Fourth Session
He remained the presenter, with Professor Wang Kelei guiding the discussion on the mean value property of heat equations and the resulting strong maximum principle, rigorously proving the uniqueness of solutions to the Cauchy problem under exponential growth constraints. Professor Wang highlighted the role of scaling transformations in simplifying complex calculations and deepening model understanding, while also exploring the theoretical basis for heat balls involving only negative time. He assigned extension exercises to encourage further exploration.
The success of the Analysis and Equations Scaling Plan is attributed to the dedication and insights of both faculty and students. Through thematic lectures and in-depth discussions, participants have not only strengthened their mathematical foundations but also gained valuable experiences in academic exploration and innovative thinking. The seminar will continue to integrate analytical methods with modern perspectives, helping students scale new heights in their academic pursuits.
(Reporters: Fang Zijin, Hu Xuehong; Photographer: Liu Xiaochun)
