TY - JOUR
T1 - W1,p estimates for elliptic homogenization problems in nonsmooth domains
AU - Shen, Zhongwei
PY - 2008
Y1 - 2008
N2 - Let Lε = -div(A(x/ε)∇), ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in ℝn, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(Lε)-1/2 are uniformly bounded on LP(Ω), where 1 < p < 3 + δ if n ≥ 3, and 1 < p < 4 + δ if n = 2. The ranges of p's are sharp. In the case of C 1 domains, we establish the uniform LP boundedness of ∇(Lε)-1/2 for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W1,p estimates for the elliptic homogenization problem Lεuε = div f in Ω, uε = 0 on ∂Ω.
AB - Let Lε = -div(A(x/ε)∇), ε > 0 be a family of second order elliptic operators with real, symmetric coefficients on a bounded Lipschitz domain Ω in ℝn, subject to the Dirichlet boundary condition. Assuming that A(x) is periodic and belongs to VMO, we show that there exists δ > 0 independent of ε such that Riesz transforms ∇(Lε)-1/2 are uniformly bounded on LP(Ω), where 1 < p < 3 + δ if n ≥ 3, and 1 < p < 4 + δ if n = 2. The ranges of p's are sharp. In the case of C 1 domains, we establish the uniform LP boundedness of ∇(Lε)-1/2 for 1 < p < ∞ and n ≥ 2. As a consequence, we obtain the uniform W1,p estimates for the elliptic homogenization problem Lεuε = div f in Ω, uε = 0 on ∂Ω.
KW - Homogenization
KW - Lipschitz domain
KW - Riesz transform
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U2 - 10.1512/iumj.2008.57.3344
DO - 10.1512/iumj.2008.57.3344
M3 - Article
AN - SCOPUS:57449118596
SN - 0022-2518
VL - 57
SP - 2283
EP - 2298
JO - Indiana University Mathematics Journal
JF - Indiana University Mathematics Journal
IS - 5
ER -