HOMOGENIZATION FOR NON-SELF-ADJOINT PERIODIC ELLIPTIC OPERATORS ON AN INFINITE CYLINDER

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Abstract

© 2017 Nikita N. Senik.We consider the problem of homogenization for non-self-adjoint second-order elliptic differential operators Aε of divergence form on L2 (ℝd1 × double-struck Td2), where d1 is positive and d2 is non-negative. The coefficients of the operator Aε are periodic in the first variable with period ε and smooth in a certain sense in the second. We show that, as ε gets small, (Aε -μ)-1 and ∇x2 (Aε -μ)-1 for an appropriate μ converge in the operator norm to, respectively, (A0 - μ)-1 and ∇x2 (A0 - μ)-1, where A0 is an operator whose coefficients depend only on x2. We also obtain an approximation for ∇x1 (Aε - μ)-1 and find the next term in the approximation for (Aε - μ)-1 . Estimates for the rates of convergence and the rates of approximation are provided and are sharp with respect to the order.
Original languageEnglish
Pages (from-to)874-898
JournalSIAM Journal on Mathematical Analysis
Volume49
Issue number2
DOIs
StatePublished - 2017

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