Jonathan Rohleder: A new approach to the hot spots conjecture

Abstract

It is a conjecture going back to J. Rauch (1974) that the hottest and coldest spots in an insulated homogeneous medium such as an insulated plate of metal should converge to the boundary, for "most" initial heat distributions, as time tends to infinity. This so-called hot spots conjecture can be phrased alternatively as follows: the eigenfunction(s) corresponding to the first non-zero eigenvalue of the Neumann Laplacian on a Euclidean domain should take its maximum and minimum on the boundary only. This has been proven to be false for certain domains with holes, but it was shown to hold for classes of simply connected or convex planar domains that either have symmetries or are thin in various senses. One of the most recent advances is the proof for all triangles given by Judge and Mondal (Annals of Math. 2020). The conjecture remains open in general for simply connected or at least convex domains. In this talk we provide a new proof of the conjecture for a class of convex planar domains that need to be neither thin nor have any symmetries.