  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.
