High-frequency homogenization of nonstationary periodic equations

2022/02/08 by Mark Dorodnyi, Dorodnyi, Mark · 1 citation
Computer Science · Mathematics · #35B27 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Boundary Problems #FOS: Mathematics

paper · pdf · doi:10.48550/arxiv.2202.03919

openalex publication_date 2022/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

We consider an elliptic differential operator A_ε = - (d)/(dx) g(x/ε) (d)/(dx) + ε^-2 V(x/ε), ε > 0, with periodic coefficients acting in L₂(ℝ). For the nonstationary Schrödinger equation with the Hamiltonian A_ε and for the hyperbolic equation with the operator A_ε, analogs of homogenization problems, related to the edges of the spectral bands of the operator A_ε, are studied (the so called high-frequency homogenization). For the solutions of the Cauchy problems for these equations with special initial data, approximations in L₂(ℝ)-norm for small ε are obtained.

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