On the Behavior of the Spectrum of a Perturbed Steklov Boundary Value Problem with a Weak Singularity

Abstract

For the Laplace equation in a bounded domain of the space $\mathbb {R}^n$, $n\geq 3$, with a boundary that contains a piece $\Gamma$ of a hyperplane $L$, we consider a boundary value problem of the Steklov type with the homogeneous Dirichlet condition on one part of the boundary and the Steklov condition on the other part. The Dirichlet condition is set outside $\Gamma$. The boundary conditions on $\Gamma$ are set in the following way. Let $R_{\delta }$ be an $(n-1)$-dimensional grid with mesh size $\delta$ in $L$, and let $A_{\varepsilon \delta }$ and $A_{2\varepsilon \delta }$ be sets of balls with centers at the vertices of the grid $R_{\delta }$ of radii $\varepsilon \delta$ and $2\varepsilon \delta$, respectively, $B_{\varepsilon \delta }=A_{2\varepsilon \delta }\setminus A_{\varepsilon \delta }$, $\widetilde {A}_{\varepsilon \delta }=\Gamma \cap A_{\varepsilon \delta }$ and $\widetilde {B}_{\varepsilon \delta }=\Gamma \cap B_{\varepsilon \delta }$, where $\varepsilon$ is a small parameter and $\delta =\delta (\varepsilon )\to 0$ as $\varepsilon \to 0$. On the piece $\Gamma$, the Dirichlet condition is posed only on the set $\widetilde {A}_{\varepsilon \delta }$. Outside this set, the spectral Steklov condition is given in which the coefficient is equal to one outside the set $\widetilde {B}_{\varepsilon \delta }$ and to $(\varepsilon \delta )^{-m}$, where $m<2$, on the set $\widetilde {B}_{\varepsilon \delta }$. For the limit (homogenized) problems and the original problem we obtain the deviations of their solutions in the norm of the Sobolev space $W_2^1$, as well as estimates for the deviations of the eigenvalues.