Operator estimates for non-periodically perforated domains: disappearance of cavities

TL;DR

This paper studies how solutions to boundary value problems in domains with many small cavities behave as the cavities shrink and become densely packed, showing they effectively disappear under certain conditions and providing convergence estimates.

Contribution

The paper establishes operator estimates for homogenization in non-periodically perforated domains with nonlinear Robin boundary conditions, demonstrating cavity disappearance and convergence rates.

Findings

Solutions converge in $W_2^1$ and $L_2$ norms to the homogenized problem.

Homogenization results hold uniformly in the $L_2$-norm of the right-hand side.

Convergence rates are estimated and their sharpness discussed.

Abstract

We consider a boundary value problem for a general second order linear equation in a perforated domain. The perforation is made by small cavities, a minimal distance between the cavities is also small. We impose minimal natural geometric conditions on the shapes of the cavities and no conditions on their distribution in the domain. On the boundaries of the cavities a nonlinear Robin condition is imposed. The sizes of the cavities and the minimal distance between them are supposed to satisfy a certain simple condition ensuring that under the homogenization the cavities disappear and we obtain a similar problem in a non-perforated domain. Our main results state the convergence of the solution of the perturbed problem to that of the homogenized one in $W_2^1$- and $L_2$-norms uniformly in $L_2$-norm of the right hand side in the equation and provide the estimates for the convergence rates. We also discuss the order sharpness of these estimates.