Elliptic Equations in Divergence Form with Partially BMO Coefficients

Dong, Hongjie; Kim, Doyoon

doi:10.1007/s00205-009-0228-7

Cited by 116 publications

(126 citation statements)

References 28 publications

Supporting

Mentioning

121

Contrasting

Order By: Relevance

“…We remark that some of our results are new even for scalar equations. In some cases, similar results were obtained in [10] and [8] for scalar elliptic and parabolic equations. In these two papers, however, a certain change of variables was used in a crucial way, which seems impossible to be applied to systems.…”

Section: Introductionsupporting

confidence: 61%

See 1 more Smart Citation

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

Dong

Kim

2010

Calc. Var.

Self Cite

View full text Add to dashboard Cite

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

show abstract

Section: Introductionsupporting

confidence: 61%

“…In this case the solution is unique up to a constant, and instead of (8.5) we have u − (u) W 1 p ( ) ≤ N g L p ( ) . Remark 8.4 As in [10], away from the lateral boundary of the domain, the coefficients A αβ are allowed to be in the partially BMO space defined by either Assumption A or Assumption A .…”

Section: Theorem 83mentioning

confidence: 99%

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

Dong

Kim

2010

Calc. Var.

Self Cite

View full text Add to dashboard Cite

show abstract

“…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. Recently, in [4,5] the authors treated divergence type second order parabolic systems in Sobolev and Orlicz spaces with A p weights and unmixed norms.…”

Section: Rubio De Franciamentioning

confidence: 99%

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

Dong

Kim

2018

Trans. Amer. Math. Soc.

Self Cite

116

155

View full text Add to dashboard Cite

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.

show abstract

“…By Theorems 8.4 and 8.5, and adapting the same arguments in the proof of Theorem 2.1 in [15] and Theorem 2.10 in [18], we can obtain the desired result. We omit the details here.…”

Section: )mentioning

confidence: 65%

Weighted Solvability for Parabolic Equations With Partially Bmo Coefficients and Its Applications

Lin

2014

J. Aust. Math. Soc.

View full text Add to dashboard Cite

We consider the weighted L p solvability for divergence and nondivergence form parabolic equations with partially bounded mean oscillation (BMO) coefficients and certain positive potentials. As an application, global regularity in Morrey spaces for divergence form parabolic operators with partially BMO coefficients on a bounded domain is established.2010 Mathematics subject classification: primary 35J55; secondary 42B25.

show abstract

Elliptic Equations in Divergence Form with Partially BMO Coefficients

Cited by 116 publications

References 28 publications

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

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

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

Weighted Solvability for Parabolic Equations With Partially Bmo Coefficients and Its Applications

Contact Info

Product

Resources

About