2024/12/29 by Andrey Piatnitski, V. A. Sloushch, Piatnitski, Andrey +5 · 1 citation
Computer Science · Engineering · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Composite Material Mechanics #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems
paper · pdf · doi:10.48550/arxiv.2412.20408
openalex publication_date 2024/12/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper deals with homogenization of self-adjoint operators in L2(\mathbb Rd) of the form (\mathbb A_\eps u) (\x) = ∫\Rd μ(\x/\eps, \y/\eps) \frac( u(\x) - u(\y) )|\x - \y|d+α d\y, where 0< α< 2, and \eps>0 is a small parameter. It is assumed that the function μ(\x,\y) is \Zd-periodic in each variable, μ(\x,\y)=μ(\y,\x) for all \x and \y, and 0< μ- \leqslant μ(\x,\y) \leqslant μ+< ∞. Under these assumptions we show that the resolvent (\mathbb A_\eps + I)-1 converges, as \eps→0, in the operator norm in L2(\Rd) to the resolvent (\mathbb A0 + I)-1 of the limit operator \mathbb A0 given by (\mathbb A0 u) (\x) = ∫\Rd μ0 \frac( u(\x) - u(\y) )|\x - \y|d+α d\y, where μ0 is the mean value of μ(\x,\y). We also show that the operator norm of the discrepancy ‖(\mathbb A_\eps + I)-1 - (\A0 + I)-1‖L2(\mathbb Rd)→ L2(\mathbb Rd) can be estimated by O(\epsα), if 0< α< 1, by O(\eps (1 + | ln \eps|)2), if α=1, and by O(\eps2- α), if 1< α< 2.