On the first eigenvalue and eigenfunction of the Laplacian with mixed
boundary conditions
On the first eigenvalue and eigenfunction of the Laplacian with mixed
boundary conditions
We consider the eigenvalue problem for the Laplacian with mixed Dirichlet and Neumann boundary conditions. For a certain class of bounded, simply connected planar domains we prove monotonicity properties of the first eigenfunction. As a consequence, we establish a variant of the hot spots conjecture for mixed boundary conditions. Moreover, …