Local Hölder continuity for fractional nonlocal equations with general growth

Byun, Sun‐Sig; Kim, Hyojin; Ok, Jihoon

doi:10.1007/s00208-022-02472-y

Cited by 21 publications

(21 citation statements)

References 39 publications

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“…To overcome this, we turn to an interpolation argument along with slightly higher fractional Sobolev regularity of a solution, which turns out to be obtained regardless of the linearity, and then run a boot strap argument in order to prove comparison estimates for fractional gradients of higher order. We believe that this method is applicable for more general nonlinear nonlocal equations with nonstandard growth [6,8,36]. We further refer to [3-5, 11-15, 18-20, 23, 24, 26-29, 32, 33, 38-40] for a further discussion of various regularity results of nonlocal problems.…”

mentioning

confidence: 99%

$L^{q}$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

Byun¹,

Kyeongbae²

2023

Preprint

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We study s-fractional p-Laplacian type equations with discontinuous kernel coefficients in divergence form to establish W s+σ,q estimates for any choice of pairs (σ, q) with q ∈ (p, ∞) and σ ∈ 0, min s p−1 , 1 − s under the assumption that the associated kernel coefficients have small BMO seminorms near the diagonal. As a consequence, we find in the literature an optimal fractional Sobolev regularity of such a non-homogeneous nonlocal equation when the right-hand side is presented by a suitable fractional operator. Our results are new even in the linear case.under a possibly discontinuous kernel coefficient A(x, y). For the classical case that p = q and σ = 0, there is a unique weak solution u ∈ W s,p (Ω) ∩ L p−1 sp (R n ) with the standard energy estimate (2.15), as follows from Lemma 2.7 below.We first discuss a motivation of the study (1.1) from the corresponding local problem when s = 1. According to the well-known elliptic theory, for a weak solution w of −div (B|Dw| p−2 Dw) = −div (|F | p−2 F ) in Ω, which comes from the Euler-Lagrange equation of the functional v → ˆΩ 1 p B|Dv| p − |F | p−2 F • Dv dx

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confidence: 99%

$L^{q}$ estimates for nonlocal p-Laplacian type equations with BMO kernel coefficients in divergence form

Byun¹,

Kyeongbae²

2023

Preprint

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“…for a constant 𝑐 ≡ 𝑐(𝑛). The rest of the proof can be done in exactly the same way as in [6,Lemma 4.1]. □…”

Section: Fractional Orlicz-sobolev Spacesmentioning

confidence: 99%

“…for a constant 𝑐 independent of 𝑡 1 and 𝑡 2 . We highlight that in the present paper we do not impose any assumption on 𝐺 other than (1.4), as in [6,34]. Indeed, we need none of (1.7) and (1.8).…”

Section: Introductionmentioning

confidence: 97%

“…In particular, in the paper [6] local boundedness and Hölder regularity of weak solutions to (1.1) are investigated under assumptions (1.3) and (1.4); see also the papers [12,22,23] based on different approaches and with additional assumptions. As a continuation of the research in [6], we aim to identify natural assumptions on g to prove Harnack inequality for weak solutions to (1.1). We mention the recent paper [20] concerning Harnack inequality for (1.1) under additional assumptions to (1.4), which are…”

Section: Introductionmentioning

confidence: 99%

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Nonlocal Harnack inequality for fractional elliptic equations with Orlicz growth

Byun

Kim

Song

2023

Bulletin of London Math Soc

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“…Such mixed local and nonlocal problems have an anisotropic feature, and they naturally link to other kinds of problems, namely nonlocal problems with nonstandard growth. Recently, the methods and results in [29,30] have been extended to nonlocal problems with various nonstandard growth conditions [7,8,9,27,32,33,42,46]; see also [14,15,16] for the extensions of the methods in [23]. Specifically, in [9] local boundedness and Hölder continuity results were proved for the nonlocal double phase problem…”

Section: Introductionmentioning

confidence: 99%