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

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 condi-tions 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.

Automated summary

In this paper, a boundary value problem for a general second-order linear equation in a perforated domain is considered. The perforation is made by small cavities with a minimal distance between them also being small. Minimal natural geometric conditions are imposed on the shapes of the cavities, while no conditions are imposed on their distribution in the domain. A nonlinear Robin condition is imposed on the boundaries of the cavities. The main results of the paper demonstrate the convergence of the solution of the perturbed problem to that of the homogenized one, providing estimates for the convergence rates in W-2(1)- and L-2-norms uniformly in the L-2-norm of the right-hand side in the equation.