The Meyers Estimates for Domains Perforated along the Boundary

Abstract

In this paper, we consider an elliptic problem in a domain perforated along the boundary. By setting a homogeneous Dirichlet condition on the boundary of the cavities and a homogeneous Neumann condition on the outer boundary of the domain, we prove higher integrability of the gradient of the solution to the problem.

On the Zaremba problem for the $p$-elliptic equation

“…Estimates of this kind, which are important in the theory of homogenization of problems with fast oscillation between boundary conditions, can be used for more accurate estimates for the convergence rate of the original solutions to that of the homogenized problem (for a similar problem in a domain perforated along the boundary, see [17]). holds for functions v ∈ W 1 p (D, F ); see below, and also see [18] and [19].…”