Global regularity for divergence form elliptic equations on quasiconvex domains

Jia, Hao; Li, Dongsheng; Wang, Lihe

doi:10.1016/j.jde.2010.08.015

Cited by 24 publications

(53 citation statements)

References 16 publications

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“…In this paper, motivated by their work in [22], we consider the quasiconvexity for the domain and extend the previous results especially in [8,22] to the nonlinear case, which is the main contribution of this work. For the detailed notion of quasiconvex domains, see the section 2.2.…”

Section: Introductionmentioning

confidence: 71%

“…Although the concept of Reifenberg flat domains itself is a very general type of domains that may have fractal boundaries, it does not contain relatively simple domains such as polygons. Motivated by this observation, the concept of quasiconvex domains is introduced as a generalization of Reifenberg in a series of recent papers [22,23,24].…”

Section: Quasiconvex Domainsmentioning

confidence: 99%

“…Then a natural question would be what conditions might be optimal for (1.5). In this regard, there have been many results including [1,7,8,9,16,22,33]. Here, we are mainly interested in minimal conditions on the boundary.…”

Section: Introductionmentioning

confidence: 99%

“…A Reifenberg domain is beyond the Lipschitz category and can be locally a topological disk. Recently, Jia, Li, and Wang brought up the concept of quasiconvex domains into the regularity theory in their papers [22,23,24]. Roughly speaking, the notion of quasiconvex domains is a generalization of Reifenberg flat domains in the sense that in all scales, their boundary can be approximated from inside and outside by two convex surfaces rather than two hyperplanes for Reifenberg flat domains.…”

Section: Introductionmentioning

confidence: 99%

“…Roughly speaking, the notion of quasiconvex domains is a generalization of Reifenberg flat domains in the sense that in all scales, their boundary can be approximated from inside and outside by two convex surfaces rather than two hyperplanes for Reifenberg flat domains. Actually, Jia, Li, and Wang in [22] investigated linear Dirichlet problem on quasiconvex domains. They proved the estimate (1.5) with p = 2.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

Byun

Kwon

2016

Journal of Mathematical Analysis and Applications

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We study the conormal derivative problem for an elliptic equation of p-Laplacian type with discontinuous coefficients in a non-smooth domain in order to look for the minimal assumptions necessary to have the nonlinear Calderón-Zygmund theory for such problem. Under the assumptions that the nonlinear operator is sufficiently close to the p-Laplacian operator in BMO semi-norm and the boundary of the domain can be locally approximated by the convex boundary, we prove that both the gradient and the associated nonhomogeneous term belong to the same L q space for every q ∈ [p, ∞). As far as the domain is concerned, our regularity assumption on the boundary is weaker than any other one reported in this direction.

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Section: Introductionmentioning

confidence: 71%

Section: Quasiconvex Domainsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

Byun

Kwon

2016

Journal of Mathematical Analysis and Applications

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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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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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Global regularity for divergence form elliptic equations on quasiconvex domains

Cited by 24 publications

References 16 publications

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

Gradient estimates for nonlinear elliptic equations with vanishing Neumann data in quasiconvex domains

Weighted variable Lorentz regularity for degenerate elliptic equations in Reifenberg domains

Gradient Weighted Norm Inequalities for Linear Elliptic Equations with Discontinuous Coefficients

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