Operator-Norm Convergence Estimates for Elliptic Homogenization Problems on Periodic Singular Structures

Abstract

For an arbitrary periodic Borel measure μ we prove order O(ε) operator-norm resolvent estimates for the solutions to scalar elliptic problems in L²(ℝ^d, dμ^ε) with ε-periodic coefficients, ε > 0. Here, μ^ε is the measure obtained by ε-scaling of μ. Our analysis includes the case of a measure absolutely continuous with respect to the standard Lebesgue measure, as well as the case of “singular” periodic structures (or “multistructures”), when μ is supported by lower-dimensional manifolds.