Boundary Hölder regularity for elliptic equations

Lian, Yuanyuan; Zhang, Kai; Li, Dongsheng; Guo, Hong

doi:10.1016/j.matpur.2020.09.012

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…The regularity theory for uniformly elliptic equations has been well developed. Taking the C k,α regularity for example, we have interior C α regularity ([27, Chapter 4], [78,79]), interior C 1,α regularity ([27, Chapter 8]), interior C 2,α regularity ( [16,50,74,75], [27,Chapter 8]), interior C k,α (k ≥ 3) regularity ( [86]), boundary C α regularity ( [88]), boundary C 1,α and C 2,α regularity ( [75,87,107]), boundary C k,α regularity ( [86]).…”

Section: Introductionmentioning

confidence: 99%

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

Lian

Zhang

2020

Journal of Differential Equations

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Section: Introductionmentioning

confidence: 99%

Boundary pointwise C1, and C2, regularity for fully nonlinear elliptic equations

Lian

Zhang

2020

Journal of Differential Equations

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“…Let us consider the function v = 1−u. Note the L ∞ -norm of v is bounded by 1, we conclude that v is bounded below by 1 2 in some neighborhood of ∂C∩ ∂B 2 by the boundary Hölder estimate (see [15,Theorem 3.6]). Combined with the Harnack inequality, v must be bounded below by a positive constant δ on ∂B 2 ∩C.…”

Section: The Foliation Structurementioning

confidence: 84%

Stability of Bernstein type theorem for the minimal surface equation

Jiang¹,

Wang²,

Zhu³

2022

Preprint

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Let Ω R n (n ≥ 2) be an unbounded convex domain. We study the minimal surface equation in Ω with boundary value given by the sum of a linear function and a bounded uniformly continuous function in R n . If Ω is not a half space, we prove that the solution is unique. If Ω is a half space, we prove that graphs of all solutions form a foliation of Ω × R. This can be viewed as a stability type theorem for Edelen-Wang's Bernstein type theorem in [9]. We also establish a comparison principle for the minimal surface equation in Ω.

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“…See also [50]. Moreover, it is known that the measure density condition of the complement of a domain is sufficient for more general linear and nonlinear nonlocal operators with measurable coefficients [34,35,39]. We point out that these conditions serve as sufficient conditions even for a wide class of local operators.…”

Section: Introductionmentioning

confidence: 87%