Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains

Lewis, John L.; Nyström, Kaj

doi:10.1515/acv.2008.005

Cited by 32 publications

(67 citation statements)

References 14 publications

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“…In [LN2] this result was extended to p = ∞ and hence to the case of infinity harmonic functions. Furthermore, in [LN1] and [LN4] a number of results concerning the boundary behavior of positive p-harmonic functions, 1 < p < ∞, in a bounded Lipschitz domain Ω ⊂ R n were proved. In particular, the boundary Harnack inequality as well as Hölder continuity for ratios of positive p-harmonic functions, 1 < p < ∞, vanishing on a portion of ∂Ω were established.…”

Section: Introductionmentioning

confidence: 99%

The boundary Harnack inequality for solutions to equations of Aronsson type in the plane

Lundström

Nyström

2011

Ann. Acad. Sci. Fenn. Math.

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Abstract. In this paper we prove a boundary Harnack inequality for positive functions which vanish continuously on a portion of the boundary of a bounded domain Ω ⊂ R 2 and which are solutions to a general equation of p-Laplace type, 1 < p < ∞. We also establish the same type of result for solutions to the Aronsson type equationConcerning Ω we only assume that ∂Ω is a quasicircle. In particular, our results generalize the boundary Harnack inequalities in [BL] and [LN2] to operators with variable coefficients.

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Section: Introductionmentioning

confidence: 99%

The boundary Harnack inequality for solutions to equations of Aronsson type in the plane

Lundström

Nyström

2011

Ann. Acad. Sci. Fenn. Math.

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“…In a subsequent paper we intend to consider the case of time-dependent weights as part of an ambition to understand the boundary behaviour of non-negative solutions to non-linear parabolic equations of p-parabolic type somehow along the lines of the elliptic theory developed in [34], [35], [38], [36]. However, already the case of time-independent weights λ(x, t) = λ(x) ∈ A 1+2/n (R n ) forces us to revisit essentially all the relevant arguments used in the corresponding context of uniformly parabolic equations.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Boundary estimates for solutions to linear degenerate parabolic equations

Nyström

Persson

Sande

2015

Journal of Differential Equations

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Let Ω ⊂ R n be a bounded NTA-domain and let Ω T = Ω × (0, T ) for some T > 0. We study the boundary behaviour of non-negative solutions to the equationWe assume that A(x, t) = {a ij (x, t)} is measurable, real, symmetric and thatfor some constant β ≥ 1 and for some non-negative and real-valued function λ = λ(x) belonging to the Muckenhoupt class A 1+2/n (R n ). Our main results include the doubling property of the associated parabolic measure and the Hölder continuity up to the boundary of quotients of non-negative solutions which vanish continuously on a portion of the boundary. Our results generalize previous results of Fabes, Kenig, Jerison, Serapioni, see [18], [19], [20], to a parabolic setting.

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“…Let L = L(n, N ) denote the collection of all (rotations of) graphs of Lipschitz functions f : R n−1 → R such that f (0) = 0 and f has Lipschitz constant at most N . In [20], Lewis and Nyström investigated the boundary behavior of solutions to the p-Laplace equation (a canonical nonlinear degenerate elliptic equation) in certain rough domains whose boundaries admit local uniform approximations by L in a Hausdorff distance sense. This class includes Lipschitz domains and domains whose boundaries are Reifenberg flat.…”

Section: Metadefinitionmentioning

confidence: 99%

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

Badger

Lewis

2015

Forum math. Sigma

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We investigate the interplay between the local and asymptotic geometry of a set A ⊆ R n and the geometry of model sets S ⊂ P(R n ), which approximate A locally uniformly on small scales. The framework for local set approximation developed in this paper unifies and extends ideas of Jones, Mattila and Vuorinen, Reifenberg, and Preiss. We indicate several applications of this framework to variational problems that arise in geometric measure theory and partial differential equations. For instance, we show that the singular part of the support of an (n − 1)-dimensional asymptotically optimally doubling measure in R n (n 4) has upper Minkowski dimension at most n − 4.

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Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains

Abstract: In this paper we prove the boundary Harnack inequality and Hölder continuity for ratios of p-harmonic functions vanishing on a portion of certain Reifenberg flat and Ahlfors regular NTA-domains. Applications are given to the p-Martin boundary problem for these domains.

Cited by 32 publications

References 14 publications

The boundary Harnack inequality for solutions to equations of Aronsson type in the plane

The boundary Harnack inequality for solutions to equations of Aronsson type in the plane

Boundary estimates for solutions to linear degenerate parabolic equations

Local Set Approximation: Mattila–vuorinen Type Sets, Reifenberg type Sets, and Tangent Sets

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