Proves that the eigenvalue index shift Ψ(d,1,Ω) between Neumann and Dirichlet Laplacians is at least C(e/2)^d for any bounded domain Ω, and the same bound holds for all k when Ω is convex.
Isoperimetric relations between Dirichlet and Neumann eigenvalues
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abstract
Inequalities between the Dirichlet and Neumann eigenvalues of the Laplacian have received much attention in the literature, but open problems abound. Here, we study the number of Neumann eigenvalues no greater than the first Dirichlet eigenvalue. Based on a combination of analytical and numerical results, we conjecture that this number is controlled by the isoperimetric ratio of the domain. This has applications to the nodal deficiency of eigenfunctions and is closely related to a long-standing conjecture of Yau on the Hausdorff measure of nodal sets.
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math.SP 1years
2026 1verdicts
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Inequalities between Dirichlet and Neumann eigenvalues in large dimensions
Proves that the eigenvalue index shift Ψ(d,1,Ω) between Neumann and Dirichlet Laplacians is at least C(e/2)^d for any bounded domain Ω, and the same bound holds for all k when Ω is convex.