Abstract:
This paper gives a systematic treatment of two methods for obtaining operator estimates: the shift method and the spectral method. Though substantially different in mathematical technique and physical motivation, these methods produce basically the same results. Besides the classical formulation of the homogenization problem, other formulations of the problem are also considered: homogenization in perforated domains, the case of an unbounded diffusion matrix, non-self-adjoint evolution equations, and higher-order elliptic operators.
Bibliography: 62 titles.
Keywords:
shift method, integrated estimate, Steklov smoothing, periodicity, problem on the cell, asymptotics of the fundamental solution, spectral method, Bloch representation of an operator, Nash–Aronson estimate.
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