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On the Poincaré inequality for periodic composite structures. (English. Russian original) Zbl 1238.28001
Proc. Steklov Inst. Math. 261, 295-297 (2008); translation from Tr. Mat. Inst. Steklova 261, 301-303 (2008).
Summary: We consider periodic composite structures characterized by a periodic Borel measure equal to the sum of at least two periodic measures. For such a composite structure, verifying the Poincaré inequality may be a difficult problem. Thus, we are interested in finding conditions under which it suffices to verify the Poincaré inequality separately for each of the simpler structure components instead of verifying it for the composite structure.
MSC:
28A12 Contents, measures, outer measures, capacities
28A05 Classes of sets (Borel fields, \(\sigma\)-rings, etc.), measurable sets, Suslin sets, analytic sets
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
74Q05 Homogenization in equilibrium problems of solid mechanics
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