Norm convergence for problems with perforation along a given manifold with nonlinear Robin condition on boundaries of cavities

TL;DR

This paper proves norm convergence of solutions for a boundary value problem with nonlinear Robin conditions on cavities along a manifold, providing homogenization results and convergence rate estimates.

Contribution

It introduces a homogenization framework for perforated domains with nonlinear boundary conditions along a manifold, including convergence rate estimates.

Findings

Solution converges in $L_2$ and $W_2^1$ norms to the homogenized solution.

Provides uniform convergence estimates in $L_2$ norm of the right-hand side.

Establishes convergence rates for the homogenization process.

Abstract

In the work we consider a boundary value problem for a second order equation with variable coefficients in a multi-dimensional domain perforated by small cavities closely spaced along a given manifold. We assume that the linear sizes of all cavities are of a same order of smallness, while their shapes and distributions are arbitrary. The boundaries of the cavities are subject to a nonlinear Robin condition. We prove that the solution of the perturbed problem converges to that of the homogenized problem in norm $L_2$ and $W_2^1$ uniformly in $L_2$-norm of the right hand side in the equation. We also establish the estimates for the convergence rates.