The homogenization of elliptic divergence-type fourth-order operators with periodic coefficients is studied in a (periodic) domain. The aim is to find an operator with constant coefficients and represent the equation through a perturbation around this operator. The resolvent is found as $L^2 \to L^2$ operator using the Neumann series for the periodic fundamental solution of biharmonic operator. Results are based on some auxiliary Lemmas suggested by Bensoussan in 1986, Zhikov in 1991, Yu. Grabovsky and G. Milton in 1998, Pastukhova in 2016. Operators of the type considered in the paper appear in the study of the elastic properties of thin plates. The choice of the operator with constant coefficients is discussed separately and chosen in an optimal way w.r.t. the spectral radius and convergence of the Neumann series and uses the known bounds for ''homogenized'' coefficients. Similar ideas are usually applied for the construction of preconditioners for iterative solvers for finite dimensional problems resulting from discretized PDEs. The method presented is similar to Cholesky factorization transferred to elliptic operators (as in references mentioned above). Furthermore, the method can be applied to non-linear problems.

翻译：研究在周期区域中具有周期系数的四阶椭圆型散度算子的均匀化问题。目标在于构造一个常系数算子，并通过该算子的摄动表示原方程。利用双调和算子周期基本解的纽曼级数，以$L^2 \to L^2$算子形式给出预解式。结果基于Bensoussan（1986）、Zhikov（1991）、Yu. Grabovsky与G. Milton（1998）以及Pastukhova（2016）提出的辅助引理。文中涉及的算子类型出现在薄板弹性性质的研究中。针对常系数算子的选取进行单独讨论，通过谱半径与纽曼级数收敛性优化选择，并利用已知的"均匀化"系数界值。类似思路常被用于构造离散偏微分方程有限维问题的迭代求解预条件器。所提方法与前述文献中椭圆算子的Cholesky分解具有相似性，且可推广至非线性问题。