Layer potential methods for elliptic homogenization problems

By Carlos E. Kenig, Zhongwei Shen. 2009

72 citations recorded by the source corpus.

In this paper we use the method of layer potentials to study L2 boundary value problems in a bounded Lipschitz domain Ω for a family of second‐order elliptic systems with rapidly oscillating periodic coefficients. Defining ${\cal L}_\varepsilon = - {\rm div}(A(\varepsilon ^{ - 1} X)\nabla )$, under the assumption that A(X) is elliptic, symmetric, periodic, and Hölder‐continuous, we establish the solvability of the L2 Dirichlet, regularity, and Neumann problems for ${\cal L}_\varepsilon (u_\varepsilon ) = 0$ in Ω with optimal estimates uniform in ε > 0. © 2010 Wiley Periodicals, Inc.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/4ade3031372b72562ba10871d90af2d7a1e9d83d

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In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Homogenization of elliptic boundary value problems in Lipschitz domains (2009, background): “This line of ideas will be fully developed in [ 23 ].”
Periodic Homogenization of Green and Neumann Functions (2014, background): “The results rely on the uniform L and Neumann function estimates obtained in [19,22] under the additional symmetry condition A D A.”
Homogenization of elliptic systems with Neumann boundary conditions (2010, unknown): “With the Lipschitz estimate in Theorem 1.2 at our disposal, Theorem 1.3 for p > 2 follows from the case p = 2 (established in [23] for Lipschitz domains), by a real-variable method originating in [9] and further developed in [29]–[31].”