Gradient estimates for elliptic systems in non-smooth domains

Byun, Sun‐Sig; Wang, Lihe

doi:10.1007/s00208-008-0207-6

Cited by 48 publications

(31 citation statements)

References 25 publications

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“…Our result is an extension of the results in [9] to the context of Newtonian fluids and weighted spaces. In [9], the authors studied…”

Section: Introductionsupporting

confidence: 61%

Weighted estimates for generalized steady Stokes systems in nonsmooth domains

Byun

2017

Journal of Mathematical Physics

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We consider a generalized steady Stokes system with discontinuous coefficients in a nonsmooth domain when the inhomogeneous term belongs to a weighted L q space for 2 < q < ∞. We prove the global weighted L q -estimates for the gradient of the weak solution and an associated pressure under the assumptions that the coefficients have small BMO (bounded mean oscillation) semi-norms and the domain is sufficiently flat in the Reifenberg sense. On the other hand, a given weight is assumed to belong to a Muckenhoupt class. Our result generalizes the global W 1.q estimate of Calderón-Zygmund with respect to the Lebesgue measure for the Stokes system in a Lipschitz domain.

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“…Our result is an extension of the results in [9] to the context of Newtonian fluids and weighted spaces. In [9], the authors studied…”

Section: Introductionsupporting

confidence: 61%

Weighted estimates for generalized steady Stokes systems in nonsmooth domains

Byun

2017

Journal of Mathematical Physics

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“…8. Note that results of this type were obtained recently in [3] for scalar equations, and in [4] for elliptic systems without lower order terms. Let us remark that the main difference of the methods is that here the boundary estimate is obtained almost immediately from the estimate in the whole space.…”

Section: Theorem 26 Letmentioning

confidence: 93%

L p solvability of divergence type parabolic and elliptic systems with partially BMO coefficients

Dong

Kim

2010

Calc. Var.

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We prove the H 1 p,q solvability of second order systems in divergence form with leading coefficients A αβ only measurable in (t, x 1 ) and having small BMO (bounded mean oscillation) semi-norms in the other variables. In addition, we assume one of the following conditions is satisfied: (i) A 11 is measurable in t and has a small BMO semi-norm in the other variables; (ii) A 11 is measurable in x 1 and has a small BMO semi-norm in the other variables. The corresponding results for the Cauchy problem and elliptic systems are also established. Some of our results are new even for scalar equations. Using the results for systems in the whole space, we obtain the solvability of systems on a half space and Lipschitz domain with either the Dirichlet boundary condition or the conormal derivative boundary condition. Mathematics Subject Classification (2000)35R05 · 35D10 · 35K51 · 35J45

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“…Then for any sufficiently small δ > 0, A αβ ij is (δ, ρ)-vanishing and Ω R is (δ, ρ)-Reifenberg flat. We should emphasize here again that δ and ρ do not depend on R. Now we consider a normalization and another scaling of functions (see Lemma 1.6 of [1]) such that …”

Section: Definitionmentioning

confidence: 99%

“…Then by adapting the proof of gradient estimate of [1] we are led to the level set estimate that for any ε > 0 there exists a δ > 0 such that…”

Section: Definitionmentioning

confidence: 99%