Geodesic
A new weighted Friedrichs-type inequality for a~perforated domain with a~sharp constant
Eurasian mathematical journal, Tome 2 (2011) no. 1, pp. 81-103

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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. 
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     title = {A new weighted {Friedrichs-type} inequality for a~perforated domain with a~sharp constant},
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G. A. Chechkin; Yu. O. Koroleva; L.-E. Persson; P. Wall. A new weighted Friedrichs-type inequality for a~perforated domain with a~sharp constant. Eurasian mathematical journal, Tome 2 (2011) no. 1, pp. 81-103. http://geodesic.mathdoc.fr/item/EMJ_2011_2_1_a3/