Operator estimates for elliptic problem with rapidly alternating Steklov boundary condition
The authors consider a boundary value problem for the Laplace equation with frequently alternating boundary conditions in a two-dimensional domain. The boundary of the domain is partitioned into many small pieces, on which the Dirichlet condition and Steklov inhomogeneous spectral condition are imposed in turns. The mentioned partition is quasi-periodic and the sizes of the Dirichlet and Steklov pieces of the boundary are such that in the limit, the homogenized condition is the Steklov one. The perturbed and the homogenized problem are reformulated in terms of some self-adjoint operators in a fractional Sobolev space on the boundary and the main result is the norm convergence of the mentioned self-adjoint operators. An estimate for the convergence rate is established. This result implies the convergence of the spectrum of the original perturbed problem to that of the homogenized ones as well as a similar convergence for the associated spectral projectors. The approach is based essentially on techniques from a recently submitted work by the reviewer et al.
scientific article; zbMATH DE number 7189419
scientific article