Degeneration of Steklov–type boundary conditions in one spectral homogenization problem

A. G. Chechkina ; V. A. Sadovnichy

Eurasian mathematical journal, Tome 6 (2015) no. 3, pp. 13-29

Résumé

We consider a singularly perturbed Steklov–type problem for the second order linear elliptic equation in a bounded two–dimensional domain. We assume that the Steklov spectral condition rapidly alternates with the homogeneous Dirichlet condition on the boundary. The alternating parts of the boundary with the Dirichlet and Steklov conditions have the same small length of order $\varepsilon$. It is proved that when the small parameter tends to zero the eigenvalues of this problem degenerate, i.e. they tend to infinity. Moreover, it is proved that the eigenvalues of the initial problem are of order $\varepsilon^{-1}$ when $\varepsilon$ tends to zero.