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Fraas, Martin; Pinchover, Yehuda

doi:10.1016/j.jde.2012.10.006

Cited by 19 publications

(3 citation statements)

References 20 publications

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“…See for instance [12] where it is proved that bounded p-harmonic functions in exterior domains has a limit at infinity. Related results can also be found in [10], [3] and [4].…”

Section: Known Resultssupporting

confidence: 61%

Decay of extremals of Morrey’s inequality

Hynd,

Larson,

Lindgren

2024

Arkiv För Matematik

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We study the decay (at infinity) of extremals of Morrey's inequality in R n . These are functions satisfying supwhere p>n and C(p, n) is the optimal constant in Morrey's inequality. We prove that if n≥2 then any extremal has a power decay of order β for any, for all functions whose first order partial derivatives belong to L p (R n ). In a series of papers (cf.[7]-[9]), Hynd and Seuffert study this inequality and prove that there is a smallest constant C >0 such that (1.1) holds and that there are extremals of this inequality. An extremal is a function for which equality is attained in (1.1). They also prove that up to translation, rotation, dilatation and multiplication by a constant, any extremal function u satisfies 1. −Δ p u=c(δ en −δ −en ) in R n for a constant c>0,

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“…See for instance [12] where it is proved that bounded p-harmonic functions in exterior domains has a limit at infinity. Related results can also be found in [10], [3] and [4].…”

Section: Known Resultssupporting

confidence: 61%

Decay of extremals of Morrey’s inequality

Hynd,

Larson,

Lindgren

2024

Arkiv För Matematik

View full text Add to dashboard Cite

show abstract

“…See for instance [11] where it is proved that bounded p-harmonic functions in exterior domains has a limit at infinity. Related results can also be found in [9], [3] and [4].…”

Section: Known Resultssupporting

confidence: 61%

Extremal functions for Morrey’s inequality in convex domains

Hynd

Lindgren

2018

Math. Ann.

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For a bounded domain Ω ⊂ R n and p > n, Morrey's inequality implies that there is c > 0 such thatfor each u belonging to the Sobolev space W 1,p 0 (Ω). We show that the ratio of any two extremal functions is constant provided that Ω is convex. We also show with concrete examples why this property fails to hold in general and verify that convexity is not a necessary condition for a domain to have this feature. As a by product, we obtain the uniqueness of an optimization problem involving the Green's function for the p-Laplacian.

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“…In the first case, if v > 0, then the limit is strictly positive; see [12,Lemma A.2]. See also the related work in [13]. As an example let Ω :…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A Liouville theorem for $$p$$ p -harmonic functions on exterior domains

2014

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ABSTRACT. We prove Liouville type theorems for p-harmonic functions on exterior domains of R d , where 1 < p < ∞ and d ≥ 2. We show that every positive p-harmonic function satisfying zero Dirichlet, Neumann or Robin boundary conditions and having zero limit as |x| tends to infinity is identically zero. In the case of zero Neumann boundary conditions, we establish that any semi-bounded p-harmonic function is constant if

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Isolated singularities of positive solutions of p-Laplacian type equations inRd

Cited by 19 publications

References 20 publications

Decay of extremals of Morrey’s inequality

Decay of extremals of Morrey’s inequality

Extremal functions for Morrey’s inequality in convex domains

A Liouville theorem for $$p$$ p -harmonic functions on exterior domains

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