Boundary Estimates in Elliptic Homogenization

By Zhongwei Shen. 2015

93 citations recorded by the source corpus.

For a family of systems of linear elasticity with rapidly oscillating periodic coefficients, we establish sharp boundary estimates with either Dirichlet or Neumann conditions, uniform down to the microscopic scale, without smoothness assumptions on the coefficients. Under additional smoothness conditions, these estimates, combined with the corresponding local estimates, lead to the full Rellich type estimates in Lipschitz domains and Lipschitz estimates in $C^{1, \alpha}$ domains. The $C^\alpha$, $W^{1,p}$, and $L^p$ estimates in $C^1$ domains for systems with VMO coefficients are also studied. The approach is based on certain estimates on convergence rates. As a bi-product, we obtain sharp $O(\varepsilon)$ error estimates in $L^q(\Omega)$ for $q=\frac{2d}{d-1}$ and a Lipschitz domain $\Omega$, with no smoothness assumption on the coefficients.

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

View Paper: https://www.semanticscholar.org/paper/e44e8a8375f404804e939344c0298d91bbf944e6

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In Applied Mathematics

In Computational Theory and Mathematics

In Mechanics of Materials

Citations in the corpus

Convergence Rates in Periodic Homogenization of Systems of Elasticity (2015, methodology): “ely in various settings. We refer the reader to [1, 7, 10] for references on earlier work in this area. More recent work on the problem of convergence rates in periodic homogenization may be found in [17, 4, 5, 13, 11, 8, 9, 12, 15, 16, 14, 6] and their references. In particular, the estimate (1.8) ”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, methodology): “4 follows the method introduced by Shen [111] for periodic coefficients; see also [77].”
Mesoscopic Higher Regularity and Subadditivity in Elliptic Homogenization (2015, background): “s the approximation step from the iteration step. Indeed, in addition to its applicability in the stochastic setting, it has yielded new results even in the periodic and almost periodic settings (cf. [3, 22]). Another idea from [4, 2] is that subadditive arguments are the way to prove quantitative h”