Homogenization of elliptic boundary value problems in Lipschitz domains

By Carlos E. Kenig, Zhongwei Shen. 2009

52 citations recorded by the source corpus.

In this paper we study the Lp boundary value problems for $${\mathcal{L}(u)=0}$$ in $${\mathbb{R}^{d+1}_+}$$ , where $${\mathcal{L}=-{\rm div} (A\nabla )}$$ is a second order elliptic operator with real and symmetric coefficients. Assume that A is periodic in xd+1 and satisfies some minimal smoothness condition in the xd+1 variable, we show that the Lp Neumann and regularity problems are uniquely solvable for 1 < p < 2 + δ. We also present a new proof of Dahlberg’s theorem on the Lp Dirichlet problem for 2 − δ < p < ∞ (Dahlberg’s original unpublished proof is given in the Appendix). As the periodic and smoothness conditions are imposed only on the xd+1 variable, these results extend directly from $${\mathbb{R}^{d+1}_+}$$ to regions above Lipschitz graphs. Consequently, by localization techniques, we obtain uniform Lp estimates for the Dirichlet, Neumann and regularity problems on bounded Lipschitz domains for a family of second order elliptic operators arising in the theory of homogenization. The results on the Neumann and regularity problems are new even for smooth domains.

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

View Paper: https://www.semanticscholar.org/paper/1dccac49f376298afafe3bc78e850bada604bec7

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In Applied Mathematics

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Citations in the corpus

Layer potential methods for elliptic homogenization problems (2009, methodology): “In [20] the approach outlined above was used to solve the L boundary value problems for elliptic equations with periodic coefficients by the method of L-harmonic measures (see Remark 1.”
Boundary Estimates in Elliptic Homogenization (2015, methodology): “2 may be used to establish uniform solvability of L(2) boundary value problems for Lε in Lipschitz domains [32, 33, 17].”
Quantitative Stochastic Homogenization and Large-Scale Regularity (2017, background): “Finer estimates on the homogenization error and two-scale expansion have been obtained more recently in [78, 76].”
Square Functions and the A∞ Property of Elliptic Measures (2016, background): “33) by well-known arguments (see [7,8,10], the discussion around (4.”
Homogenization of elliptic systems with Neumann boundary conditions (2010, unknown): “We mention that (6.3) as well as the uniform L 2 Rellich estimate for the regularity of Dirichlet problem (6.4) was established by Kenig and Shen in [23] under the assumption that Ω is Lipschitz, A ∈ Λ( μ, λ, τ ), and A = A ∗ (also see [22] for the case of the elliptic equation).”
Uniform Regularity Estimates in Parabolic Homogenization (2013, methodology): “Also see related work in [3, 5, 4, 6, 12, 31, 22, 23, 20, 17, 32] for various uniform estimates in elliptic homogenization.”
Periodic Homogenization of Green and Neumann Functions (2014, background): “In the case that is Lipschitz, the map is uniformly bounded for 1 < p < 2C ı if m D 1 [21] and for p close to 2 if m > 1 [22].”