'Optimal Shapes of Level Sets' | David Jerison | MIT 2020

Описание: MIT 2020: Calculus of Variations, Homogenization and Disorder — As the month of August made way fro September, members, students, and frequent collaborators of the Simons Collaboration on Localization of Waves met via Zoom to discuss their latest research. With a host of different speakers presenting a wide array of research, four days at the end of the summer were filled with productive math and physics discussion.

Abstract: Level surfaces of eigenfunctions, free boundaries, and isoperimetric surfaces divide space and partition energy in optimal ways. The calculus of variations allows us to see strong parallels among the methods used to understand all of these types of surfaces. We begin by explaining the Hot Spots Conjecture of Jeff Rauch concerning the shape of the first non-constant Neumann eigenfunction in convex domains. One approach to proving it is to establish clean separation of level sets, a kind of Harnack inequality. So far we are only able to prove analogous results for free boundaries and for isoperimetric surfaces, not level surfaces of eigenfunctions. For free boundaries, the key step is understanding how a convex cone divides itself into two cones so as to minimize a suitable energy, an extension of the classical Friedland-Hayman inequality from all of space to convex cones. The work presented is mostly joint with one or another combination of Thomas Beck, Guy David, and Sarah Raynor.

Slides: https://drive.google.com/file/d/1MDje...

MIT 2020 webpage: z.umn.edu/MIT2020
Simons Collaboration on Localization of Waves website: cse.umn.edu/WAVE