A new approach to the hot spots conjecture

We introduce a new variational principle for the study of eigenvalues and eigenfunctions of the Laplacians with Neumann and Dirichlet boundary conditions on planar domains. In contrast to the classical variational principles, its minimizers are gradients of eigenfunctions instead of the eigenfunctions themselves. This variational principle enables us to give an elementary analytic proof of the famous hot spots conjecture for the class of so-called lip domains. More specifically, we show that each eigenfunction corresponding to the lowest positive eigenvalue of the Neumann Laplacian on such a domain is strictly monotonous along two mutually orthogonal directions. In particular, its maximum and minimum may only be located on the boundary.