2023/09/27 by T. V. Anoop, Anoop, T. V., В. И. Бобков +3 · 2 citations
Computer Science · Mathematics · #34L15 #35P15 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2309.15558
openalex publication_date 2023/09/27 · openalex created_date 2023/09/30 · openalex updated_date 2026/07/28
Let τk(Ω) be the k-th eigenvalue of the Laplace operator in a bounded domain Ω of the form Ωout ∖ Bα under the Neumann boundary condition on ∂ Ωout and the Robin boundary condition with parameter h ∈ (-∞,+∞] on the sphere ∂ Bα of radius α>0 centered at the origin, the limiting case h=+∞ being understood as the Dirichlet boundary condition on ∂ Bα. In the case h>0, it is known that the first eigenvalue τ1(Ω) does not exceed τ1(Bβ∖ Bα), where β>0 is chosen such that |Ω| = |Bβ∖ Bα|, which can be regarded as a reverse Faber-Krahn type inequality. We establish this result for any h ∈ (-∞,+∞]. Moreover, we provide related estimates for higher eigenvalues under additional geometric assumptions on Ω, which can be seen as Szegő-Weinberger type inequalities. A few counterexamples to the obtained inequalities for domains violating imposed geometric assumptions are given. As auxiliary information, we investigate shapes of eigenfunctions associated with several eigenvalues τi(Bβ∖ Bα) and show that they are nonradial at least for all positive and all sufficiently negative h when i ∈ \2,…,N+2\. At the same time, we give numerical evidence that, in the planar case N=2, already second eigenfunctions can be radial for some h<0. The latter fact provides a simple counterexample to the Payne nodal line conjecture in the case of the mixed boundary conditions.