An end-point global gradient weighted estimate for quasilinear equations in non-smooth domains

Adimurthi, Karthik; Phuc, Nguyen Cong

doi:10.1007/s00229-015-0804-0

Cited by 17 publications

(21 citation statements)

References 33 publications

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“…One popular approach is the approximation method pioneered by Caffarelli and Peral in [10] that avoids the use of singular integral theory directly but rather studies the integrability of gradient of solutions as a function of the deviation of the coefficients from constant coefficients. That method has been successfully implemented in [7,8] with the use of Hardy-Littlewood maximal function to treat divergence form data on irregular domains; see also [3,4,31,32,33,26,25,9] where the method is used in different function spaces such as weighted Lebesgue and Lorentz spaces, Lorentz-Morrey spaces, and Orlicz spaces. Another approach that employs a local version of the sharp maximal function of Fefferman and Stein was implemented in [29,30] (see also the earlier work of T. Iwaniec [22]).…”

Section: Discussion On the Approachmentioning

confidence: 99%

“…The work [36] (see also [31,32,33]) yields weighted estimate (1.4) for q > 2 and for weights w ∈ A q/2 . In [4], the first and last named authors worked out the end-point case q = 2 for weights w ∈ A 1 . Those weighted estimates also hold true for equations with general nonlinear structures such as those that are modeled after the p-Laplacian.…”

Section: Introductionmentioning

confidence: 99%

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Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

2018

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Local and global weighted norm estimates involving Muckenhoupt weights are obtained for gradient of solutions to linear elliptic Dirichlet boundary value problems in divergence form over a Lipschitz domain Ω. The gradient estimates are obtained in weighted Lebesgue and Lorentz spaces, which also yield estimates in Lorentz-Morrey spaces as well as Hölder continuity of solutions. The significance of the work lies on its applicability to very weak solutions (that belong to W 1,p 0 (Ω) for some p > 1 but not necessarily in W 1,2 0 (Ω)) to inhomogeneous equations with coefficients that may have discontinuities but have a small mean oscillation. The domain is assumed to have a Lipschitz boundary with small Lipschitz constant and as such allows corners. The approach implemented makes use of localized sharp maximal function estimates as well as known regularity estimates for very weak solutions to the associated homogeneous equations. The estimates are optimal in the sense that they coincide with classical weighted gradient estimates in the event the coefficients are continuous and the domain has smooth boundary.

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Section: Discussion On the Approachmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

2018

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“…The purpose of this paper is twofold: Firstly, we bridge the gap between the estimates (1.3) and (1.4) to obtain the relation 5) provided 1 ≤ q − ≤ q(·) ≤ q + < ∞, i.e., we allow q − = 1 and q(·) is log-Hölder continuous (we assume the same structure conditions on the nonlinearity A as in [10]). This represents an end point case of the estimate (1.4).…”

Section: P(·)mentioning

confidence: 99%

Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

Adimurthi

Byun

Park

2018

Journal of Functional Analysis

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In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the p(x)-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of [10], the estimates obtained were strictly above the natural exponent and hence there was a gap between the natural energy estimates and estimates above p(x), see (1.3) and (1.4). Here, we bridge this gap to obtain the end point case of the estimates obtained in [10], see (1.5). In order to do this, we have to obtain significantly improved a priori estimates below p(x), which is the main contribution of this paper. We also improve upon the previous results by obtaining the estimates for a larger class of domains than what was considered in the literature.

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“…The study of regularity estimates in various function spaces for linear or non-linear elliptic equations (or systems) in non-smooth domains is one of the most interesting and important topics in partial differential equations (see, for instance, [1,7,13,17,26,29,32,33,43,44] for the linear case and [2,3,10,18,19,21,51,52] for the non-linear case). Furthermore, it is well known that the global regularity estimates for solutions to elliptic boundary problems depend not only on the structure of equations and the properties of the right-hand side datum and the coefficients appearing in equations, but also on the smooth property or the geometric property of the boundary of domains (see, for instance, [1,7,16,21,29,33,44,48,51]).…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 1.7(i). Then there exists a positive constant δ 0 ∈ (0, ∞), depending only on n, p, θ and the Lipschitz constant of Ω, such that, if A satisfies the (δ, R)-BMO condition for some δ ∈ (0, δ 0 ) and R ∈ (0, ∞) or A ∈ VMO(R n ), then, for any weak solution u ∈ W 1,2 (Ω) of the Robin problem (R) 2…”

Section: Introductionmentioning

confidence: 99%

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

Yang

Chang

Yang

et al. 2020

Journal of Differential Equations

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Let n ≥ 2 and Ω be a bounded Lipschitz domain in R n . In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Ω. More precisely, let p ∈ (n/(n − 1), ∞). Using a real-variable argument, the authors obtain two necessary and sufficient conditions for W 1,p estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse Hölder inequality with exponent p or weighted W 1,q estimates of solutions with q ∈ (n/(n − 1), p] and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second order elliptic equations of divergence form with small BMO coefficients, respectively, on bounded Lipschitz domains, C 1 domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when n = 3 in case of bounded C 1 domains, also gives an alternative correct proof of some know result. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak-)Orlicz spaces and variable Lebesgue spaces.

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An end-point global gradient weighted estimate for quasilinear equations in non-smooth domains

Cited by 17 publications

References 33 publications

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

Sharp gradient estimates for quasilinear elliptic equations with p(x) growth on nonsmooth domains

Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

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