A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results

Bhattacharya, Tilak

doi:10.1155/2007/78029

Cited by 5 publications

(4 citation statements)

References 12 publications

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“…When p=∞ the proof is similar, but then the comparison argument uses cones (which are ∞-harmonic) in place of the function φ p (x) defined on [4, page 286], and therefore the exterior ball condition is not needed. See also [14] for the case p=∞.…”

Section: Lemma 24 Assume That P∈(1 ∞]mentioning

confidence: 99%

Estimates of $p$-harmonic functions in planar sectors

Lundström

Singh

2023

Arkiv För Matematik

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Suppose that p∈(1, ∞], ν ∈[1/2, ∞), Sν = (x 1 , x 2 )∈R 2 \{(0, 0)}:|φ|< π 2ν , where φ is the polar angle of (x 1 , x 2 ). Let R>0 and ωp(x) be the p-harmonic measure of ∂B(0, R)∩Sν at x with respect to B(0, R)∩Sν . We prove that there exists a constant C such thatwhenever x∈B(0, R)∩S 2ν and where the exponent k(ν, p) is given explicitly as a function of ν and p. Using this estimate we derive local growth estimates for p-sub-and p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmén-Lindelöf theorem for p-subharmonic functions in the unbounded sector Sν . Moreover, if p=∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in R n . Finally, when ν ∈(1/2, ∞) and p∈(1, ∞) we prove uniqueness (modulo normalization) of positive p-harmonic functions in Sν vanishing on ∂Sν .

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Section: Lemma 24 Assume That P∈(1 ∞]mentioning

confidence: 99%

Estimates of $p$-harmonic functions in planar sectors

Lundström

Singh

2023

Arkiv För Matematik

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“…See also [37,38,39] for similar as well as stronger estimates in more general geometries. When p = ∞ the proof is similar, but then the comparison argument uses cones (which are ∞-harmonic) in place of the function φ p (x) defined on [8, page 286], and therefore the exterior ball condition is not needed see [14] for the case p = ∞.…”

Section: Lemma 26mentioning

confidence: 99%

Estimates of $p$-harmonic functions in planar sectors

Lundström¹,

Singh²

2021

Preprint

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, where φ is the polar angle of (x, y). Let R > 0 and ω p (x) be the p-harmonic measure of ∂B(0, R) ∩ S v at x with respect to B(0, R) ∩ S v . We prove that there exists a constant C such thatwhere the exponent k(v, p) is given explicitly as a function of v and p. Using this estimate we derive local growth estimates for p-suband p-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of p-harmonic measure we also derive a sharp Phragmen-Lindelöf theorem for p-subharmonic functions in the unbounded sector S v . Moreover, if p = ∞ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in R n . Finally, when v ∈ (1/2, ∞) and p ∈ (1, ∞) we prove a uniqueness result (modulo normalization) for positive p-harmonic functions in S v vanishing on ∂S v .

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“…We use now the standard Harnack inequality for infinity harmonic functions in Θ (see e.g. [5]) to derive the existence of c This implies that for a fixed σ 0 ∈ S, one has…”

Section: Proof Of Theorem Dmentioning

confidence: 99%

“…The two functions ψ α and ω α depend only on the variable φ ∈ (0, α] and are unique in the class of rotanionnaly invariant solutions up to multiplication by constants. This study reduced to an ordinary differential equation which has been already treated by T. Bhattacharya in [4] and [5]. But for the sake of completeness and for some related problems we present it in Section 3 of the present paper.…”

Section: Introductionmentioning

confidence: 99%

Separable infinity harmonic functions in cones

2018

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We study the existence of separable infinity harmonic functions in any cone of R N vanishing on its boundary under the form u(r, σ) = r −β ψ(σ). We prove that such solutions exist, the spherical part ψ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N −1 and that the exponents β = β + > 0 and β = β − < 0 are uniquely determined if the domain is smooth. We extend some of our results to non-smooth domains.

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A Boundary Harnack Principle for Infinity-Laplacian and Some Related Results

Cited by 5 publications

References 12 publications

Estimates of $p$-harmonic functions in planar sectors

Estimates of $p$-harmonic functions in planar sectors

Estimates of $p$-harmonic functions in planar sectors

Separable infinity harmonic functions in cones

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