Sobolev Spaces on Metric Measure Spaces

Heinonen, Juha; Koskela, Pekka; Shanmugalingam, Nageswari; Tyson, Jeremy T.

doi:10.1017/cbo9781316135914

Cited by 361 publications

(314 citation statements)

References 34 publications

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“…For Ω = X, with X locally compact and under global assumptions, similar extension results appear in Aikawa-Shanmugalingam [1,Proposition 7.1] and Heinonen-Koskela-Shanmugalingam-Tyson [22,Lemma 8.2.3]. Theorem 1.2 makes it possible to study functions u on X using properties known to hold for their extensionsû on X.…”

Section: Introductionmentioning

confidence: 67%

“…We also assume that 1 < p < ∞, although the results in Sections 2 and 3 also hold if p = 1. Proofs of the results in this section can be found in the monographs Björn-Björn [3] and Heinonen-Koskela-Shanmugalingam-Tyson [22].…”

Section: Upper Gradients and Newtonian Spacesmentioning

confidence: 83%

“…Much of analysis on metric spaces has been done assuming global doubling and global Poincaré inequalities, which for instance are assumed in the monographs Haj lasz-Koskela [18], Björn-Björn [3] and Heinonen-Koskela-Shanmugalingam-Tyson [22]. For wider applicability we study properties that hold under more local assumptions.…”

Section: Introductionmentioning

confidence: 99%

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Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Björn¹,

Björn²

2019

Journal of Differential Equations

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Let X be a noncomplete metric space satisfying the usual (local) assumptions of a doubling property and a Poincaré inequality. We study extensions of Newtonian Sobolev functions to the completion X of X and use them to obtain several results on X itself, in particular concerning minimal weak upper gradients, Lebesgue points, quasicontinuity, regularity properties of the capacity and better Poincaré inequalities. We also provide a discussion about possible applications of the completions and extension results to pharmonic functions on noncomplete spaces and show by examples that this is a rather delicate issue opening for various interpretations and new investigations.

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Section: Introductionmentioning

confidence: 67%

Section: Upper Gradients and Newtonian Spacesmentioning

confidence: 83%

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Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Björn¹,

Björn²

2019

Journal of Differential Equations

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“…Later we will impose further requirements on the space and on the measure. We will keep the discussion short, see the monographs Björn-Björn [11] and Heinonen-Koskela-Shanmugalingam-Tyson [35] for proofs, further discussion, and references on the topics in this section.…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

Björn

Hansevi

2019

Analysis and Geometry in Metric Spaces

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The theory of boundary regularity for p-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a p-Poincaré inequality, 1 < p < ∞. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.

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“…For every sufficiently large λ ≥ 1 there exists a measurable set E ⊂ B(0,3 4 ) with |E| < ε λ 2 and such that the restriction of u to D \ E is λ-Lipschitz. This essentially follows from the proof of[16, Theorem 8.2.1]. For the convenience of the reader, we provide the proof.Proof.…”

mentioning

confidence: 97%

Spaces with almost Euclidean Dehn function

Wenger

2019

Math. Ann.

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We prove that any proper, geodesic metric space whose Dehn function grows asymptotically like the Euclidean one has asymptotic cones which are non-positively curved in the sense of Alexandrov, thus are CAT(0). This is new already in the setting of Riemannian manifolds and establishes in particular the borderline case of a result about the sharp isoperimetric constant which implies Gromov hyperbolicity. Our result moreover provides a large scale analog of a recent result of Lytchak and the author which characterizes proper CAT(0) in terms of the growth of the Dehn function at all scales. We finally obtain a generalization of this result of Lytchak and the author. Namely, we show that if the Dehn function of a proper, geodesic metric space is sufficiently close to the Euclidean Dehn function up to some scale then the space is not far (in a suitable sense) from being CAT(0) up to that scale.Date: November 9, 2018. 1991 Mathematics Subject Classification. 53C23, 20F65, 49Q05.

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Sobolev Spaces on Metric Measure Spaces

Cited by 361 publications

References 34 publications

Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Poincaré inequalities and Newtonian Sobolev functions on noncomplete metric spaces

Boundary Regularity for p-Harmonic Functions and Solutions of Obstacle Problems on Unbounded Sets in Metric Spaces

Spaces with almost Euclidean Dehn function

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