The limiting equation for Neumann Laplacians on shrinking domains

Abstract

Let ${Omega_{epsilon} }_{0 < epsilon le1}$ be an indexed family of connected open sets in ${mathbb R}^2$, that shrinks to a tree $Gamma$ as $epsilon$ approaches zero. Let $H_{Omega_{epsilon}}$ be the Neumann Laplacian and $f_{epsilon}$ be the restriction of an $L^2(Omega_1)$ function to $Omega_{epsilon} $. For $z in {mathbb C}Backslash [0, infty)$, set $u_{epsilon} = (H_{Omega_{epsilon}} - z)^{-1}f_{epsilon} $. Under the assumption that all the edges of $Gamma$ are line segments, and some additional conditions on $Omega_{epsilon}$, we show that the limit function $u_0 = lim_{epsilono 0} u_{epsilon}$ satisfies a second-order ordinary differential equation on $Gamma$ with Kirchhoff boundary conditions on each vertex of $Gamma $.

Keywords

• Neumann Laplacian
• tree
• shrinking domains.