Calculus of Variations and Geometric Measure Theory

D. Bucur - A. Henrot - M. Michetti

Asymptotic behaviour of the Steklov problem on dumbbell domains

created by bucur on 10 Jul 2020
modified by michetti1 on 17 Feb 2022


Published Paper

Inserted: 10 jul 2020
Last Updated: 17 feb 2022

Journal: Comm. Partial Differential Equations
Volume: 46
Number: 2
Pages: 362-393
Year: 2021

ArXiv: 2007.04844 PDF


We analyse the asymptotic behaviour of the eigenvalues and eigenvectors of a Steklov problem in a dumbbell domain consisting of two Lipschitz sets connected by a thin tube with vanishing width. All the eigenvalues are collapsing to zero, the speed being driven by some power of the width which multiplies the eigenvalues of a one dimensional problem. In two dimensions of the space, the behaviour is fundamentally different from the third or higher dimensions and the limit problems are of different nature. This phenomenon is due to the fact that only in dimension two the boundary of the tube has not vanishing surface measure.