Tim Schmatzler: Monotonicity property for certain eigenfunctions on lip-domains

Abstract: The ‘hot spots conjecture’ due to Rauch states that the second eigenfunction of the Neumann Laplacian on ‘nice’ domains must take its maximum at the boundary. J. Rohleder gave a purely analytic proof of this fact for planar so-called lip-domains by showing that a second Neumann eigenfunctions is monotone in two orthogonal coordinate directions. We extend these methods to certain operators with coefficients, and suggest an application concerning the nodal line certain Robin Laplace eigenfunctions on rectangles.

This is based on joint work (in progress) with Jonathan Rohleder.