Article
Ufa Mathematical Journal
Volume 3, Number 3, pp. 122-134
On estimate of eigenfunctions to the Steklov--type problem with small parameter in case of limit spectrum degeneration
Chechkina A.G., Sadovnichii V.A.
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We consider a Steklov--type problem with rapidly alternating boundary conditions (Dirichlet and Steklov) in a bounded two-dimensional domain. The parts of the boundary with the Dirichlet boundary condition have the length of order $\varepsilon$ and they alternate with parts with the length of the same order, having the Steklov condition. We prove that for a sufficiently small $\varepsilon$ the normalized eigenfunctions satisfy the Friedrichs--type inequality with the constant of order $\varepsilon$, and moreover they converge to zero as $\varepsilon$ tends to zero.