On the symmetry and monotonicity of Morrey extremals

Hynd, Ryan; Seuffert, Francis

doi:10.3934/cpaa.2020238

Cited by 5 publications

(2 citation statements)

References 24 publications

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“…it coincides with that of the whole space (see Corollary 4.3 below, for example). Some studies on such a constant and its extremals have been done recently by Hynd and Seuffert in a series of papers (see [23,24] and [25]), but the exact value of the sharp constant is still unknown. We show in Corollary 4.3 that lim p→∞ m p (Ω)…”

Section: Remark 13 (Comparison With Previous Resultsmentioning

confidence: 99%

Sobolev embeddings and distance functions

Brasco¹,

Prinari²,

Chiara³

2023

Preprint

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On a general open set of the euclidean space, we study the relation between the embedding of the homogeneous Sobolev space D 1,p 0 into L q and the summability properties of the distance function. We prove that in the superconformal case (i.e. when p is larger than the dimension) these two facts are equivalent, while in the subconformal and conformal cases (i.e. when p is less than or equal to the dimension) we construct counterexamples to this equivalence. In turn, our analysis permits to study the asymptotic behaviour of the positive solution of the Lane-Emden equation for the p−Laplacian with sub-homogeneous right-hand side, as the exponent p diverges to ∞. The case of first eigenfunctions of the p−Laplacian is included, as well. As particular cases of our analysis, we retrieve some well-known convergence results, under optimal assumptions on the open sets. We also give some new geometric estimates for generalized principal frequencies.

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Section: Remark 13 (Comparison With Previous Resultsmentioning

confidence: 99%

Sobolev embeddings and distance functions

Brasco¹,

Prinari²,

Chiara³

2023

Preprint

View full text Add to dashboard Cite

show abstract

“…In a series of papers (cf. [8,6,7]), 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).…”

Section: Introductionmentioning

confidence: 99%

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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On Morrey's inequality in Sobolev-Slobodeckiĭ spaces

Brasco,

Prinari,

2024

Journal of Functional Analysis

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On the symmetry and monotonicity of Morrey extremals

Cited by 5 publications

References 24 publications

Sobolev embeddings and distance functions

Sobolev embeddings and distance functions

Extremal functions for Morrey’s inequality in convex domains

On Morrey's inequality in Sobolev-Slobodeckiĭ spaces

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