Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

Dong, Hongjie; Kim, Doyoon

doi:10.1016/j.jfa.2011.08.001

Cited by 77 publications

(126 citation statements)

References 32 publications

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“…The proof follows those of Corollary 4.2, Lemma 4.6, Corollary 7.6, and Lemma 7.7 of [18] by using the interior and boundary H m q estimates, i.e., Theorems 2.2 and 2.4 in the same paper, instead of the H m 2 estimates.…”

Section: Higher Order Systems In Divergence Form In Reifenberg Domainsmentioning

confidence: 83%

“…The coefficients are, roughly speaking, locally measurable in one direction and have small mean oscillations with respect to the other directions in small balls or cylinders. If no weights and unmixed norms are assumed, the results in Section 6 have been developed in [18]. Concerning the second order equations/systems case, see, for instance, [17,19,21,8,6] and references therein.…”

Section: Rubio De Franciamentioning

confidence: 99%

“…For this approach, see, for instance, [40,42,20]. When deriving desired mean oscillation estimates, we take full advantage of the existence and uniqueness results as well as unmixed L pestimates for second and higher order elliptic and parabolic equations/systems in Sobolev spaces without weights proved in [20,36,16,18]. As to divergence type systems in Reifenberg flat domains, here we prove a priori L p,q -estimates using sharp functions, whereas previously a level set type argument in the spirit of [9] is used.…”

Section: Rubio De Franciamentioning

confidence: 99%

“…Indeed, this was mainly due to the lack of the corresponding Fefferman-Stein theorem. For equations/systems in Reifenberg flat domain and the level set type argument, see, for instance, [7,18]. Utilizing sharp functions makes it possible to derive mixednorm estimates from mean oscillation estimates via the Fefferman-Stein theorem with mixed norms.…”

Section: Rubio De Franciamentioning

confidence: 99%

“…We follow the lines in the proof of Proposition 7.10 in [18], where Ω = R d + and q = 2. As mentioned in Remark 5.8, we assume that the coefficients are infinitely differentiable.…”

Section: Higher Order Systems In Divergence Form In Reifenberg Domainsmentioning

confidence: 99%

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On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Dong

Kim

2018

Trans. Amer. Math. Soc.

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Abstract. We prove generalized Fefferman-Stein type theorems on sharp functions with Ap weights in spaces of homogeneous type with either finite or infinite underlying measure. We then apply these results to establish mixednorm weighted Lp-estimates for elliptic and parabolic equations/systems with (partially) BMO coefficients in regular or irregular domains. IntroductionThe objective of this paper is two-fold. The first is to present a few generalized versions of the Fefferman-Stein theorem on sharp functions. One of our main theorems in this direction proveswhere (X , µ) is a space of homogeneous type, w is a Muckenhoupt weight, f # dy is the sharp function of f using a dyadic filtration of partitions of X , and L p,q (X , w dµ) is a mixed norm. A space X of homogeneous type is equipped with a quasi-distance and a doubling measure µ. A brief description of spaces of homogeneous type is given in Section 2. For more discussions, see [45,44,11]. If X is the product of two spaces (X 1 , µ 1 ) and (X 2 , µ 2 ) of homogeneous type, w = w 1 (x ′ )w 2 (x ′′ ), and µ(x) = µ 1 (x ′ )µ 2 (x ′′ ), where x ′ ∈ X 1 and x ′′ ∈ X 2 , then the L p,q (X , w dµ) norm is defined asSee (2.13) for a precise definition of the mixed norm. If X is the Euclidean space R d , µ is the Lebesgue measure on R d , w ≡ 1, and p = q, the inequality (1.1) is the celebrated Fefferman-Stein theorem on sharp functions. See, for instance, [26,49], where the measure of the underlying space is clearly infinite. Here we deal with both the cases of a finite measure µ(X ) < ∞ and an infinite measure µ(X ) = ∞. We also present a different form of the Fefferman-Stein theorem. See Theorems 2.3 and 2.4 and Corollaries 2.6 and 2.7.The second objective, which is in fact a motivation of writing this paper, is to apply the generalized versions of the Fefferman-Stein theorem to establishing 2010 Mathematics Subject Classification. 35R05, 42B37, 35B45, 35K25, 35J48. Key words and phrases. sharp/maximal functions, elliptic and parabolic equations, weighted Sobolev spaces, measurable coefficients.H.

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Section: Higher Order Systems In Divergence Form In Reifenberg Domainsmentioning

confidence: 83%

Section: Rubio De Franciamentioning

confidence: 99%

Section: Rubio De Franciamentioning

confidence: 99%

Section: Rubio De Franciamentioning

confidence: 99%

“…We follow the lines in the proof of Proposition 7.10 in [18], where Ω = R d + and q = 2. As mentioned in Remark 5.8, we assume that the coefficients are infinitely differentiable.…”

Section: Higher Order Systems In Divergence Form In Reifenberg Domainsmentioning

confidence: 99%

See 3 more Smart Citations

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Dong

Kim

2018

Trans. Amer. Math. Soc.

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116

155

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Gradient estimates in generalized Morrey spaces for parabolic operators

Byun

Softova

2015

Mathematische Nachrichten

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We obtain global regularity in generalized Morrey spaces for the gradient of the weak solutions to divergence form linear parabolic operators with measurable data. Assuming partial BMO smallness of the coefficients and Reifenberg flatness of the boundary of the underlying domain, we develop a Calderón-Zygmund type theory for such operators. Problems like the considered here arise in the modeling of composite materials and in the mechanics of membranes and films of simple nonhomogeneous materials which form a linear laminated medium.

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Weighted variable Lorentz regularity for degenerate elliptic equations in Reifenberg domains

Zhang

Yang

2021

Math Methods in App Sciences

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Let w be a Muckenhoupt A 2 (R n ) weight and Ω a Reifenberg flat domain in R n . Assume that q ∈ (0, ∞] and p(•) : Ω → (1, ∞) is a variable exponent satisfying the log-Hölder continuous condition. In this article, the authors investigate the weighted variable Lorentz L p(•), q (Ω, w) regularity of the weak solutions of second-order degenerate elliptic equations in divergence form with Dirichlet boundary condition, under the assumption that the degenerate coefficients belong to weighted BMO spaces with small norms.

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Higher order elliptic and parabolic systems with variably partially BMO coefficients in regular and irregular domains

Cited by 77 publications

References 32 publications

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

On $L_p$-estimates for elliptic and parabolic equations with $A_p$ weights

Gradient estimates in generalized Morrey spaces for parabolic operators

Weighted variable Lorentz regularity for degenerate elliptic equations in Reifenberg domains

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