On the L p -Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

Dong, Hongjie; Kim, Doyoon

doi:10.1007/s00205-010-0345-3

Cited by 111 publications

(162 citation statements)

References 28 publications

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“…We shall generalize some results in [20] to the case of mixed-norm Lebesgue spaces with A p weights.…”

Section: Higher Order Parabolic Systems In Non-divergence Form With Bmentioning

confidence: 89%

“…The special case of the lemma when q = 2 was proved in [20,Lemma 3], which was derived from Lemma 2 there. The latter still holds with q in place of 2 thanks to the W 1,2m q estimates established in the same paper.…”

Section: Higher Order Parabolic Systems In Non-divergence Form With Bmentioning

confidence: 99%

“…Coefficients in this class (called BMO x coefficients) are less regular than those having vanishing mean oscillations (VMO). See [20] and references therein for a discussion about BMO and VMO coefficients. Here we generalize the results in [20], where no weights and unmixed norms are considered.…”

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].…”

Section: Rubio De Franciamentioning

confidence: 99%

“…We mainly follow the arguments in the previous subsection and Sections 7-9 of [20]. For a function g defined on a subset D in R d+1 , we set…”

Section: 2mentioning

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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“…We shall generalize some results in [20] to the case of mixed-norm Lebesgue spaces with A p weights.…”

Section: Higher Order Parabolic Systems In Non-divergence Form With Bmentioning

confidence: 89%

Section: Higher Order Parabolic Systems In Non-divergence Form With Bmentioning

confidence: 99%

Section: Rubio De Franciamentioning

confidence: 99%

Section: Rubio De Franciamentioning

confidence: 99%

“…We mainly follow the arguments in the previous subsection and Sections 7-9 of [20]. For a function g defined on a subset D in R d+1 , we set…”

Section: 2mentioning

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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120

155

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Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

Lo,

Rodrigues

2024

Mathematische Nachrichten

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We consider the quasilinear diffusion problem of for an open set , , for , and any . Here, denotes an operator which may involve the distributional Riesz fractional gradient of order , with , the classical gradient or/and nonlocal derivatives , with . We show global existence results for various quasilinear diffusion systems in nondivergence form for linear elliptic operators , including classical elliptic systems, anisotropic fractional equations and systems, and anisotropic local and nonlocal operators of the following type: for coercive, invertible matrices and suitable vectorial functions .

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Lorentz estimates for quasilinear elliptic double obstacle problems involving a Schrödinger term

Nguyen

Tran

2021

Math Methods in App Sciences

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Our goal in this article is to study the global Lorentz estimates for gradient of weak solutions to p‐Laplace double obstacle problems involving the Schrödinger term: −normalΔpu+𝕍false|ufalse|p−2u with bound constraints ψ1 ≤ u ≤ ψ2 in nonsmooth domains. This problem has its own interest in mathematics, engineering, physics, and other branches of science. Our approach makes a novel connection between the study of Calderón‐Zygmund theory for nonlinear Schrödinger type equations and variational inequalities for double obstacle problems.

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On the L p -Solvability of Higher Order Parabolic and Elliptic Systems with BMO Coefficients

Cited by 111 publications

References 28 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

Global existence for nonlocal quasilinear diffusion systems in nonisotropic nondivergence form

Lorentz estimates for quasilinear elliptic double obstacle problems involving a Schrödinger term

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