Pointwise estimate for the solutions of elliptic equations in periodic perforated domains is concerned. Let ε denote the size ratio of the period of a periodic perforated domain to the whole domain. It is known that even if the given functions of the elliptic equations are bounded uniformly in ε, the C1,α norm and the W2,p norm of the elliptic solutions may not be bounded uniformly in ε. It is also known that when ε closes to 0, the elliptic solutions in the periodic perforated domains approach a solution of some homogenized elliptic equation. In this work, the Hölder uniform bound in ε and the Lipschitz uniform bound in ε for the elliptic solutions in perforated domains are proved. The L∞ and the Lipschitz convergence estimates for the difference between the elliptic solutions in the perforated domains and the solution of the homogenized elliptic equation are derived.
- Homogenized elliptic equation
- Periodic perforated domains
- Two-phase media