Sugata Mondal: Hot spots for a class of Lip-1 domains

Abstract: Hot spots conjecture, due J. Rauch, says that the global extremum of a second Neumann eigenfunction of a domain cannot be attained at an interior point of the domain. I will briefly recall the history of the problem leading to the recent resolution claimed by OpenAI for simply connected domains in the Euclidean plane. In higher dimensions, it seems thigs are more complicated. Last year, J. Dios Pont announced a result claiming that hot spots conjecture is not true for general convex domains in high dimensions. In this talk, I will focus on a result originally due to Atar-Burdzy, for a class of Lip-1 domains (domains in the plane bounded by the graphs of two Lipschitz curves with Lipschitz constant at most 1), where they showed that hot spots conjecture is true. Later, with my collaborator Chirs Judge, we gave an independent proof of this result under certain conditions (and around the same time, J. Rohleder also gave an independent proof). The main result of this talk would be a geometric analogue of this result in higher dimensions.