Weakly Singular Steklov Condition in the Multidimensional Case

In an n-dimensional (n > 3) domain, we consider a Steklov-type problem with rapidly changing conditions (the Steklov condition alternates with the homogeneous Dirichlet condition). The coefficient in the Steklov condition is a rapidly oscillating function depending on a small parameter epsilon and having the order O(1) outside small spherical layer inclusions and the order O((epsilon delta)(-m)) inside them. These inclusions have an O(epsilon delta) diameter and lie at a distance of O(delta) from each other, where delta = delta(epsilon) -> 0. In the case m < 2 (weak singularity), the rate of convergence of solutions to the original problem as the small parameter tends to zero is estimated.

Automated summary

This problem involves a rapidly changing Steklov problem in an n-dimensional domain. The condition alternates between Steklov condition and homogeneous Dirichlet condition. The coefficient in the Steklov condition is a rapidly oscillating function and has different orders inside and outside the inclusions. The convergence rate of the solution to the original problem is estimated when the small parameter tends to zero in the case of weak singularity.