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

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.


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
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