This paper focuses on the asymptotic behavior of the eigenvalues of the classical Steklov problem in a thin multidomain as it becomes increasingly narrow. The thin multidomain consists of two vertical cylinders, one positioned atop the other. We show that the eigenvalues tend to zero as the domain becomes thinner. In this case, the limiting eigenvalue problem in the upper cylinder corresponds to a one-dimensional eigenvalue problem, while the limiting problem in the lower cylinder is defined in an -dimensional ball. We conclude our investigation by considering the case in which the thinness of the upper and lower cylinders is equal.
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