2025/06/08 by Andrey Piatnitski, V. A. Sloushch, Piatnitski, Andrey +5
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.2506.07176
openalex publication_date 2025/06/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The paper studies homogenization problem for a bounded in L2(\mathbb Rd) convolution type operator \mathbb A_\eps, \eps >0, of the form (\mathbb A_\eps u) (\x) = \eps-d-2 ∫\Rd a((\x-\y)/\eps) μ(\x/\eps, \y/\eps) ( u(\x) - u(\y) ) d\y. It is assumed that a(\x) is a non-negative function from L1(\Rd), and μ(\x,\y) is a periodic in \x and \y function such that 0< μ- \leqslant μ(\x,\y) \leqslant μ+< ∞. No symmetry assumption on a(⋅) and μ(⋅) is imposed, so the operator \mathbb A_\eps need not be self-adjoint. Under the assumption that the moments Mk = ∫\Rd |\x|k a(\x) d\x, k=1,2,3, are finite we obtain, for small \eps>0, sharp in order approximation of the resolvent (\mathbb A_\eps + I)-1 in the operator norm in L2(\mathbb Rd), the discrepancy being of order O(\eps). The approximation is given by an operator of the form (\mathbb A0 + \eps-1 ⟨ \boldsymbolα,∇ ⟩ + I)-1 multiplied on the right by a periodic function q0(\x/\eps); here \mathbb A0 = - divg0 ∇ is the effective operator, and \boldsymbolα is a constant vector.