Mathematical Problems in Elasticity and Homogenization

By O. Oleinik, A. Shamaev, G. Yosifian. 2012

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Convergence Rates in Periodic Homogenization of Systems of Elasticity (2015, background): “W e refer the reader to [1, 7, 10] for references on earlier work in this area.”
Boundary Estimates in Elliptic Homogenization (2015, neutral): “re Ab= baαβ ij is the matrix of eﬀective coeﬃcients with baαβ ij = − ˆ Y n aαβ ij +a αγ ik ∂ ∂yk χγβ j o . (2.2) It is well known that the constant matrix Absatisﬁes the elasticity condition (1.2) [32, 23]. Deﬁne bαβ ij (y) = a αβ ij +a αγ ik ∂ ∂yk χγβ j −baαβ ij . (2.3) By the deﬁnition of Aband”
Strain gradient and generalized continua obtained by homogenizing frame lattices (2018, background): “This problem has been widely studied in static or dynamic cases, for conduction or elasticity problems, when the cell is made of a material with varying properties [18], [64], [7], [62].”
Homogenization of elliptic systems with Neumann boundary conditions (2010, positive): “e have overcome this diﬃculty, in the presence of symmetry, thanks to the Rellich estimates obtained in [21, 22]. Neumann boundary conditions are important in applications of homogenization (see e.g. [8, 18, 26, 27]). The uniform estimates we establish in this paper can be used to study convergence ”