A counterexample to the “hot spots” conjecture. (English) Zbl 0919.35094

Burdzy, Krzysztof; Werner, Wendelin

Ann. Math. (2) 149, No. 1, 309-317 (1999).

Summary: We construct a counterexample to the “hot spots” conjecture; there exists a bounded connected planar domain (with two holes) such that the second eigenvalue of the Laplacian in that domain with Neumann boundary conditions is simple and such that the corresponding eigenfunction attains its strict maximum at an interior point of that domain.

MSC:

35P15	Estimates of eigenvalues in context of PDEs
35J05	Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35B05	Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Keywords:

second eigenvalue of the Laplacian