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

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.

KEYWORDS:

• Perforated domain
• non-periodic perforation
• operator estimates
• convergence rate

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35B27
• 35B40

Acknowledgments

The author thanks the referees for careful reading the original version of the paper and numerous useful comments, which allowed the author to improve the original version of the paper.

Data availability

Not applicable in the manuscript as no datasets were generated or analysed during the current study.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The work is supported by the Czech Science Foundation within the project 22-18739S.