The Courant-Herrmann conjecture

Abstract

The Courant-Herrmann Conjecture (CHC) concerns the sign properties of combinations of the Dirichlet eigenfunctions of elliptic pde’s, the most important of which is the Helmholtz equation Δu + λρu = 0 for D ∈ R↑N. If the eigenvalues are ordered increasingly, CHC states that the nodal set of a combination v = SIGMA ciui (1=<i<=n) of the first n eigenfunctions, divides D into no more than n sign domains in which v has one sign. The conjecture is classically known to hold for N = 1, we conjecture that it is true for rectangular boxes in R↑N(N ≥ 2), but show that it is false in general.