Parabolic and Elliptic Equations with VMO Coefficients

Крылов, Н. В.

doi:10.1080/03605300600781626

Cited by 245 publications

(335 citation statements)

References 14 publications

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“…A priori estimates for the elliptic system in (5.4) defined in the whole spaces and on a half space are derived by using the corresponding estimates in Theorems 5.2 and 5.4 for the parabolic system and the argument, for instance, in the proof of [40,Theorem 2.6]. The key idea is that one can view an elliptic system as a steady state parabolic system.…”

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

confidence: 99%

See 1 more Smart Citation

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

Dong

Kim

2018

Trans. Amer. Math. Soc.

116

155

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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 Parabolic Systems In Non-divergence Form With Bmentioning

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%

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

Dong

Kim

2018

Trans. Amer. Math. Soc.

116

155

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“…This paper is a natural continuation of our previous investigations [9], [8]. By combining the techniques from these articles we investigate parabolic equations of type u t + a jk (t, x)u x j x k + b j (t, x)u x j + c(t, x)u = f (1.1)…”

Section: Introductionmentioning

confidence: 99%

“…Before that the Sobolev space theory was established for some other types of discontinuities [11], [10], [4], [16] (see also [6] ( [7]) for a modern approach covering p = 2 in the elliptic (parabolic) case). Solvability theory for discontinuous coefficients is important not only from pure theoretical point of view but also from the point of view of applications, for instance, to random diffusion processes, see, for instance, [17], [9]. Observe that the class of equations with VMO coefficients and the class of equations with discontinuities treated in [11], [4], [16], [6], [7] have no common members apart from the equations with just continuous coefficients.…”

Section: Introductionmentioning

confidence: 99%

Parabolic Equations with Measurable Coefficients

2007

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“…We note in passing that the proofs of Propositions 2.2 and 2.5 are of analytic nature (relying for proving Prop. 2.2 on results from [26,27,29] about parabolic equations with non-smooth coefficients), whereas those of Propositions 2.6 and 2.7 are mostly probabilistic, involving tools from large deviations theory and stochastic analysis. More precisely, the local large deviations upper bound of Prop.…”

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confidence: 99%

Large Deviations for Diffusions Interacting Through Their Ranks

Dembo

Shkolnikov

Varadhan

et al. 2016

Comm Pure Appl Math

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Abstract. We prove a Large Deviations Principle (ldp) for systems of diffusions (particles) interacting through their ranks, when the number of particles tends to infinity. We show that the limiting particle density is given by the unique solution of the appropriate McKean-Vlasov equation and that the corresponding cumulative distribution function evolves according to a non-degenerate generalized porous medium equation with convection. The large deviations rate function is provided in explicit form. This is the first instance of a ldp for interacting diffusions, where the interaction occurs both through the drift and the diffusion coefficients and where the rate function can be given explicitly. In the course of the proof, we obtain new regularity results for tilted versions of such generalized porous medium equation.

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Parabolic and Elliptic Equations with VMO Coefficients

Cited by 245 publications

References 14 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

Parabolic Equations with Measurable Coefficients

Large Deviations for Diffusions Interacting Through Their Ranks

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