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PDEs & Applications Seminar

Friday, October 11th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Alexandre Girouard (Université Laval)

Title: Singular perturbations, homogenization and conformal approximation in spectral geometry: tools for eigenvalue optimization and flexibility

Abstract: In this talk, I will discuss recent advances in spectral geometry that use techniques from homogenization theory—originally developed in industrial mathematics—and conformal perturbations to address spectral optimization problems. These methods allow an elliptic eigenvalue problem (P) to be approximated as the limit of a family of related problems (P_t), often of a different character. The solutions of (P_t) are carefully controlled to approximate those of (P) in the limit. This approach yields sharp isoperimetric bounds for eigenvalues of the Dirichlet-to-Neumann map on planar domains and shows the instability of maximizers in well-known spectral optimization problems. I will also discuss a simpler singular perturbation: excision of thin tubular neighborhoods around submanifolds, which in some cases offers more direct proofs of similar results.