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Periodic homogenization of Green and Neumann functions. (English) Zbl 1300.35030

The very interesting paper under review deals with the asymptotic behaviours of the Green and Neumann functions for a family of uniformly elliptic operators with rapidly oscillating coefficients of the type \[ -\text{div\,}\big(A(x/\varepsilon)\nabla\big)=-\dfrac{\partial}{\partial x_i}\left[ a^{\alpha\beta}_{ij}\left(\frac{x}{\varepsilon}\right)\dfrac{\partial}{\partial x_j}\right],\quad \varepsilon>0, \] with Hölder continuous coefficients. By using Dirichlet and Neumann correctors, the authors obtain asymptotic expansions of Poisson kernels and the Dirichlet-to-Neumann maps. Moreover, optimal convergence rates in \(L^p\) and \(W^{1,p}\) are established for solutions with Dirichlet or Neumann boundary conditions.

MSC:

35J47 Second-order elliptic systems
35J08 Green’s functions for elliptic equations
35C20 Asymptotic expansions of solutions to PDEs
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
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