A self-adjoint strongly elliptic second-order differential operator A_ε on L₂(ℝd;ℂⁿ) is considered. It is assumed that the coefficients of A_ε are periodic and depend on x/ε, where ε > 0 is a small parameter. Approximations for the operators cos(A ^1/2_ε τ) and A ^1/2_ε sin(A ^1/2_ε τ) in the norm of operators from the Sobolev space H^s(ℝ^d;ℂⁿ) to L₂(ℝ^d;ℂⁿ) (for appropriate s) are obtained. Approximation with a corrector for the operator A ^1/2_ε sin(A ^1/2_ε τ) in the (H^s → H¹)-norm is also obtained. The question about the sharpness of the results with respect to the norm type and with respect to the dependence of the estimates on is studied. The results are applied to study the behavior of the solutions of the Cauchy problem for the hyperbolic equation ∂ ²_τ u_ε = − Aεu_ε.

中文翻译：

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双曲方程均化的算子误差估计

甲自伴随强烈椭圆二阶微分算子甲_ε上大号₂（ℝ d ;ℂ ^Ñ）被认为。假定的系数甲_ε是周期性的，并且依赖于X / ε，其中ε > 0是一个小的参数。近似为运营商COS（甲^1/2 _ε τ）和甲^1/2 _ε SIN（甲^1/2 _ε τ从索伯列夫空间算的范数）ħ^小号（ℝ ^d ;ℂ ^Ñ ）到大号₂（ℝ ^d ;ℂ ^Ñ）（对于适当的小号）获得。逼近为操作一个校正甲^1/2 _ε SIN（甲^1/2 _ε τ在）（ħ^小号→ ħ ¹）范数也被获得。研究了关于规范类型和估计依赖关系的结果的清晰度的问题。结果应用于研究Cauchy问题为双曲线方程解的行为∂ ² _τ Ù _ε = - Aε ù _ε。