Abstract
In this paper, we consider periodic soft inclusions T ε with periodicity ε, where the solution, u ε , satisfies semi-linear elliptic equations of non-divergence in \({\Omega_{\epsilon}=\Omega\setminus \overline{T}_\epsilon}\) with Neumann data on \({\partial T^{\mathfrak a}}\). The difficulty lies in the non-divergence structure of the operator where the standard energy method, which is based on the divergence theorem, cannot be applied. The main object is to develop a viscosity method to find the homogenized equation satisfied by the limit of u ε , referred to as u, as ε approaches to zero. We introduce the concept of a compatibility condition between the equation and the Neumann condition on the boundary for the existence of uniformly bounded periodic first correctors. The concept of a second corrector is then developed to show that the limit, u, is the viscosity solution of a homogenized equation.
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Communicated by G. Dal Maso
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Lee, Ka., Yoo, M. The Viscosity Method for the Homogenization of Soft Inclusions. Arch Rational Mech Anal 206, 297–332 (2012). https://doi.org/10.1007/s00205-012-0533-4
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DOI: https://doi.org/10.1007/s00205-012-0533-4