Geometric PDE and Applied Analysis Seminar (September 27, 2024)

Talk 1: 10:00-11:00

Speaker: Prof. Tatsuya Miura (Kyoto University)

Title: Variational stabilization of degenerate p-elasticae

Abstract: In this talk we discuss a new stabilization phenomenon induced by degenerate diffusion in the context of pinned planar p-elasticae. More precisely, in the non-degenerate regime (including the classical case of Euler's elastica) there are no local minimizers other than unique global minimizers, but in the degenerate regime there emerges an uncountable family of local minimizers with unbounded energy.

Talk 2: 11:15-12:15

Speaker: Prof. Hiroyoshi Mitake (University of Tokyo)

Title: Quantitative homogenization of state-constraint Hamilton–Jacobi equations on perforated domains and applications

Abstract: We first summarize the recent development of qualitative and quantitative homogenization theory for periodic Hamilton–Jacobi equations. Then, we focus on a more recent topic of state-constraint Hamilton–Jacobi equations on perforated domains. We establish the optimal convergence rate in the convex setting. Moreover, we then consider a dilute situation in which the holes’ diameter is much smaller than the microscopic scale. Finally, a homogenization problem with domain defects where some holes are missing is analyzed. This is a joint work with Yuxi Han, Wenjia Jing, Hung V. Tran.