A new weighted Friedrichs-type inequality for a perforated domain with a sharp constant. (English) Zbl 1234.35027

Summary: We derive a new three-dimensional Hardy-type inequality for a cube for the class of functions from the Sobolev space \(H^1\) having zero trace on small holes distributed periodically along the boundary. The proof is based on a careful analysis of the asymptotic expansion of the first eigenvalue of a related spectral problem and the best constant of the corresponding Friedrichs-type inequality.


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
39A10 Additive difference equations
39A70 Difference operators
39B62 Functional inequalities, including subadditivity, convexity, etc.
41A44 Best constants in approximation theory
45A05 Linear integral equations
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals