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

Asymptotic behaviour of the Steklov problem on dumbbell domains