Approximate Conditions and Passage to the Limit in Sobolev Spaces over Thin and Composite Structures

Approximate properties of variable measures compose a base of the two-scale convergence method with respect to a variable measure, which is used in averaging on periodic thin and composite structures. The paper gives a survey of methods for verifying approximate properties. The main focus is on studying composite measures, which have been little studied. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 16, Partial Differential Equations, 2004.     

On the Korn Inequalities on Thin Periodic Frames

Korn-type inequalities for thin periodic structures of period and width h() with h() 0 are presented. Periodic meshes, three-dimensional road structures, and three-dimensional box structures are considered. A particular attention is paid to structures with the so-called critical width when 0$$ " align="middle" border="0"> .     

Gamma–closure of some classes of nonstandard convex integrands

Within the framework of the theory of Γ-convergence for convex functionals with nonstandard coercivity and growth conditions, we study the inheritance of some properties (for example, strict convexity, differentiablility, and Δ₂-property) of integrands under taking the Γ-closure. We focus on power integrands |ξ| ^p(x) with variable exponents. The results obtained are also of interest in the case of the Γ-convergence theory for standard functionals. Bibliography: 15 titles.