On the eigenvalues of the biharmonic operator with Neumann boundary conditions on a thin set

Abstract

Let Ω be a bounded domain in ℝ 2 with smooth boundary 𝜕Ω, and let 𝜔 ℎ be the set of points in Ω whose distance from the boundary is smaller than ℎ. We prove that the eigenvalues of the biharmonic operator on 𝜔 ℎ with Neumann boundary conditions converge to the eigenvalues of a limiting problem in the form of a system of differential equations on 𝜕Ω.M S C 2 0 2 0 35J40 (primary), 35B25, 35J35, 35P20 (secondary) INTRODUCTION AND STATEMENT OF THE MAIN RESULTLet Ω be a bounded domain in ℝ 2 with smooth boundary 𝜕Ω. For ℎ > 0, we define the domain 𝜔 ℎ asWe consider the Neumann eigenvalue problem for the biharmonic operator in 𝜔 ℎ , namely𝜕 2 𝜈𝜈 𝑢 ℎ = 0 , on 𝜕𝜔 ℎ , div 𝜕𝜔 ℎ (𝐷 2 𝑢 ℎ ⋅ 𝜈) 𝜕𝜔 ℎ + 𝜕 𝜈 Δ𝑢 ℎ = 0 , on 𝜕𝜔 ℎ ,