Saito, Yoshimi2020-01-072020-01-072000-04-26Saito, Y. (2000). The limiting equation for Neumann Laplacians on shrinking domains. <i>Electronic Journal of Differential Equations, 2000</i>(31), pp. 1-25.1072-6691https://hdl.handle.net/10877/9138Let {Ω∊}0<∊ ≤1 be an indexed family of connected open sets in ℝ², that shrinks to a tree Γ as ∊ approaches zero. Let HΩ∊ be the Neumann Laplacian and ƒ∊ be the restriction of an L²(Ω₁) function to Ω∊. For z ∈ ℂ\ [0, ∞), set u∊ = (HΩ∊ - z)-1 ƒ∊. Under the assumption that all the edges of Γ are line segments, and some additional conditions on Ω∊, we show that the limit function u0 = lim∊→0u∊ satisfies a second-order ordinary differential equation on Γ with Kirchhoff boundary conditions on each vertex of Γ.Text25 pages1 file (.pdf)enAttribution 4.0 InternationalNeumann LaplacianTreeShrinking domainsArticle