2021/06/25 by Barbara Brandolini, Brandolini, Barbara, Francesco Chiacchio +3 · 1 citation
Computer Science · Mathematics · #35J92 #35P15 #35P30 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2106.13903
openalex publication_date 2021/06/25 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
We prove explicit and sharp eigenvalue estimates for Neumann p-Laplace eigenvalues in domains that admit a representation in Fermi coordinates. More precisely, if γ denotes a non-closed curve in ℝ2 symmetric with respect to the y-axis, let D⊂ ℝ2 denote the domain of points that lie on one side of γ and within a prescribed distance δ(s) from γ(s) (here s denotes the arc length parameter for γ). Write μ1odd(D) for the lowest nonzero eigenvalue of the Neumann p-Laplacian with an eigenfunction that is odd with respect to the y-axis. For all p>1, we provide a lower bound on μ1odd(D) when the distance function δ and the signed curvature k of γ satisfy certain geometric constraints. In the linear case (p=2), we establish sufficient conditions to guarantee μ1odd(D)=μ1(D). We finally study the asymptotics of μ1(D) as the distance function tends to zero. We show that in the limit, the eigenvalues converge to the lowest nonzero eigenvalue of a weighted one-dimensional Neumann p-Laplace problem.