Nonlinear gradient estimates for elliptic equations in quasiconvex domains

Byun, Sun‐Sig; Kwon, Hun; So, Hyoungsuk; Wang, Lihe

doi:10.1007/s00526-015-0830-5

Cited by 10 publications

(4 citation statements)

References 22 publications

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“…Obviously, A(w) = |w| p−2 w, and A(w) = a(w)w (a(w) is defined in [4]), satisfy (1.3)-(1.5). Therefore we call (1.1) the p-Laplacian type equation.…”

Section: )mentioning

confidence: 96%

“…[1,[3][4][5]) mainly focus on zero boundary data (ϕ = 0). In [4], Cianchi and Maz'ya considered the Lipschitz regularity with right hand side terms in some Lorentz spaces. And their results hold on domains with some curvature restrictions.…”

Section: )mentioning

confidence: 99%

“…And very recently, Byun and his collaborators (cf. [3]) showed the boundary regularity of (1.1) on the quasiconvex domain (ϕ = 0). In this work, we consider (1.1) with general ϕ ∈ W 1,q .…”

Section: )mentioning

confidence: 99%

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Quasilinear elliptic equations on convex domains

Jia

Wang

2016

Journal of Mathematical Analysis and Applications

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We show global regularity for p-Laplacian type elliptic equations on convex domains with W 1,q (q ≥ p > 1) boundary data. The maximal function, Vitali covering lemma and approximation (or compactness method) are the main techniques in the proof. To derive the approximation lemma, we need W 1,∞ estimates for the corresponding homogeneous equations, where the convexity of the domain plays an important role.

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“…Obviously, A(w) = |w| p−2 w, and A(w) = a(w)w (a(w) is defined in [4]), satisfy (1.3)-(1.5). Therefore we call (1.1) the p-Laplacian type equation.…”

Section: )mentioning

confidence: 96%

Section: )mentioning

confidence: 99%

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Quasilinear elliptic equations on convex domains

Jia

Wang

2016

Journal of Mathematical Analysis and Applications

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“…We mention that another well-known boundary perturbation approach applied to Reifenberg flat domains, originating from Caffarelli and Peral [10], was developed by Wang and Byun in [5] (more application may be found in, e.g., [6,7,8,9,22]), which uses compactness and a proof of contradiction (as a result, the dependence of constants cannot be specified). Their approach was also used in [16,17,4] in the setting of quasiconvex domains. Our new approach in this paper, quite different from theirs, is straightforward and do not use compactness or the proof of contradiction.…”

Section: 3mentioning

confidence: 99%

Weak maximum principle for biharmonic equations in quasiconvex Lipschitz domains

Zhuge

2019

Preprint

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In dimension two or three, the weak maximum principal for biharmonic equation is valid in any bounded Lipschitz domains. In higher dimensions (greater than three), it was only known that the weak maximum principle holds in convex domains or C 1 domains, and may fail in general Lipschitz domains. In this paper, we prove the weak maximum principle in higher dimensions in quasiconvex Lipschitz domains, which is a sharp condition in some sense and recovers both convex and C 1 domains.

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