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.

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.

This paper is devoted to a comparison between the normalized first (non-trivial) Neumann eigenvalue |\Omega| \mu_1(\Omega) for a Lipschitz open set \Omega in the plane and the normalized first (non-trivial) Steklov eigenvalue P(\Omega) \sigma_1(\Omega) . More precisely, we study the ratio F(\Omega):=|\Omega| \mu_1(\Omega)/P(\Omega) \sigma_1(\Omega) . We prove that this ratio can take arbitrarily small or large values if we do not put any restriction on the class of sets \Omega . Then we restrict ourselves to the class of plane convex domains for which we get explicit bounds. We also study the case of thin convex domains for which we give more precise bounds. The paper finishes with the plot of the corresponding Blaschke–Santaló diagrams (x,y)=(|\Omega| \mu_1(\Omega), P(\Omega) \sigma_1(\Omega) ) .

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