This paper deals with the homogenization of a mixed boundary value problem for the Laplace operator in a domain with locally periodic oscillating boundary. The Neumann condition is prescribed on the oscillating part of the boundary, and the Dirichlet condition on a separate part. It is shown that the homogenization result holds in the sense of weak L 2 convergence of the solutions and their flows, under natural hypothesis on the regularity of the domain. The strong L 2 convergence of average preserving extensions of the solutions and their flows is also considered.
We consider the homogenization of a singularly perturbed selfadjoint fourth order elliptic equation with locally periodic coefficients, stated in a bounded domain. We impose Dirichlet boundary conditions on the boundary of the domain. The presence of large parameters in the lower order terms and the dependence of the coefficients on the slow variable give rise to the effect of localization of the eigenfunctions. We show that the jth eigenfunction can be approximated by a rescaled function that is constructed in terms of the jth eigenfunction of fourth or second order order effective operators with constant coefficients, depending on the large parameters.
Abstract. We give an example of a relation between local and effective properties for elastic structures, up to geometric constants. The model considered is a periodic structure with isotropic and homogeneous local elasticity tensor in planar linear elasticity. The corresponding physical model is a flat two dimensional body with holes as, for example, a perforated plate.
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