On Stein's extension operator preserving Sobolev–Morrey spaces

Lamberti, Pier Domenico; Violo, Ivan Yuri

doi:10.1002/mana.201700480

Cited by 8 publications

(10 citation statements)

References 13 publications

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“…Below, we choose to compare their extensions. The extensions we seek rely on the Stein universal extension operator (see [47] and [33,29]). We recall the following from from [47].…”

Section: Then We Use First Inequality and The Continuous Embeddingmentioning

confidence: 99%

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On the Polygonal Faber-Krahn Inequality

Bogosel¹,

Bucur²

2022

Preprint

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It has been conjectured by Pólya and Szegö seventy years ago that the planar set which minimizes the first eigenvalue of the Dirichlet-Laplace operator among polygons with n sides and fixed area is the regular polygon. Despite its apparent simplicity, this result has only been proved for triangles and quadrilaterals. In this paper we prove that for each n ≥ 5 the proof of the conjecture can be reduced to a finite number of certified numerical computations. Moreover, the local minimality of the regular polygon can be reduced to a single numerical computation. For n = 5, 6, 7, 8 we perform this computation and certify the numerical approximation by finite elements, up to machine errors.

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“…Below, we choose to compare their extensions. The extensions we seek rely on the Stein universal extension operator (see [47] and [33,29]). We recall the following from from [47].…”

Section: Then We Use First Inequality and The Continuous Embeddingmentioning

confidence: 99%

“…Since the boundary of P is described in the same charts as the boundary of the regular polygon, we use the explicit formula of the extension operator. We refer the reader to [47, Theorem 5, page 181] (see also [33,29]), where the explicit construction is given.…”

Section: Then We Use First Inequality and The Continuous Embeddingmentioning

confidence: 99%

On the Polygonal Faber-Krahn Inequality

Bogosel¹,

Bucur²

2022

Preprint

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Section: Morrey Spacesmentioning

confidence: 99%

“…We also refer the interested readers to [23] for a different construction of the extension operator. Note that in [23], the authors only considered the higher order Morrey-Sobolev spaces M p,s k (B 1 ). However, the proof works with minor changes (replacing the L p estimates by corresponding weak L p * estimates) for the higher order weak Morrey-Sobolev spaces M p,s k, * (B 1 ).…”

Section: Morrey Spacesmentioning

confidence: 99%

$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

Guo¹,

Wang²,

Xiang³

2021

Preprint

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We establish an optimal L p -regularity theory for solutions to fourth order elliptic systems with antisymmetric potentials in all supercritical dimensions n ≥ 5:where and D, E, Ω, G satisfy the growth condition (GC-4), under the smallness condition of a critical scale invariant norm of ∇u and ∇ 2 u. This system was brought into lights from the study of regularity of (stationary) biharmonic maps between manifolds by Lamm-Rivière, Struwe, and Wang.In particular, our results improve Struwe's Hölder regularity theorem to any Hölder exponent α ∈ (0, 1) when f ≡ 0, and have applications to both approximate biharmonic maps and heat flow of biharmonic maps. As a by-product of the techniques, we also extend the L p -regularity theory of harmonic maps by Moser to Rivière-Struwe's second order elliptic systems with antisymmetric potentials under the growth condition (GC-2) in all dimensions, which confirms an expectation by Sharp.

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“…With reference to the problem of the extension of Sobolev-Morrey spaces, besides [21], we would also like to quote the papers [27], [30], [41].…”

Section: Introductionmentioning

confidence: 99%

REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

Lamberti¹,

Vespri²

2019

EMJ

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We discuss a few old results concerning embedding theorems for Campanato and Sobolev-Morrey spaces adapting the formulations to the case of domains of class C 0,γ , and we present more recent results concerning the extension of functions from Sobolev-Morrey spaces defined on those domains. As a corollary of the extension theorem we obtain an embedding theorem for Sobolev-Morrey spaces on arbitrary C 0,γ domains.

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On Stein's extension operator preserving Sobolev–Morrey spaces

Cited by 8 publications

References 13 publications

On the Polygonal Faber-Krahn Inequality

On the Polygonal Faber-Krahn Inequality

$L^p$-regularity for fourth order elliptic systems with antisymmetric potentials in higher dimensions

REMARKS ON SOBOLEV-MORREY-CAMPANATO SPACES DEFINED ON C_{0;\gamma} DOMAINS

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