Regularity of quasi-minimizers on metric spaces

Kinnunen, Juha; Shanmugalingam, Nageswari

doi:10.1007/s002290100193

Cited by 179 publications

(237 citation statements)

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“…The paper [B] has a better capacitary version of this inequality, but for our purposes it suffices to consider the more easily proved version below (see [KSh,Lemma 2.1]).…”

Section: Sobolev Inequalitiesmentioning

confidence: 99%

Polar sets on metric spaces

Kinnunen

Shanmugalingam

2005

Trans. Amer. Math. Soc.

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Abstract. We show that if X is a proper metric measure space equipped with a doubling measure supporting a Poincaré inequality, then subsets of X with zero p-capacity are precisely the p-polar sets; that is, a relatively compact subset of a domain in X is of zero p-capacity if and only if there exists a p-superharmonic function whose set of singularities contains the given set. In addition, we prove that if X is a p-hyperbolic metric space, then the p-superharmonic function can be required to be p-superharmonic on the entire space X. We also study the the following question: If a set is of zero pcapacity, does there exist a p-superharmonic function whose set of singularities is precisely the given set? IntroductionSets of zero capacity in potential theory play the same role as sets of zero measure in the study of L p -spaces. Given a metric measure space X, functions in the equivalence classes in the Sobolev space N 1,p (X) agree up to sets of zero p-capacity. The purpose of this note is to characterize sets of zero p-capacity in terms of the singular sets of p-superharmonic functions.A polar set of a metric space is a set E ⊂ X so that there exists a p-superharmonic function which takes on the value of infinity at every point in E. When X is a Euclidean space and E is a subset of a domain in this Euclidean space with zero p-capacity, then there exists a p-superharmonic function on the domain taking on the value of infinity at each point of E. On the other hand, it is known that polar sets have zero p-capacity. Hence, the collection of all sets of zero p-capacity in X and the collection of all polar sets in X are one and the same collection (see [HKM] and the references therein). We extend this result to the setting of metric measure spaces that are equipped with a doubling measure and supporting a (1, q)-Poincaré inequality for some q with 1 ≤ q < p. For recent results on analysis and potential theory on metric spaces we refer to [B] Kil2] is not applicable in the generality considered in this paper. However, we do follow the general idea of the proof given

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“…The paper [B] has a better capacitary version of this inequality, but for our purposes it suffices to consider the more easily proved version below (see [KSh,Lemma 2.1]).…”

Section: Sobolev Inequalitiesmentioning

confidence: 99%

Polar sets on metric spaces

Kinnunen

Shanmugalingam

2005

Trans. Amer. Math. Soc.

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“…Giaquinta-Giusti [14] proved several fundamental properties for quasiminimizers including the interior regularity result that a quasiminimizer can be modified on a set of zero measure so that it becomes Hölder continuous. These results were extended to metric spaces by Kinnunen-Shanmugalingam [24].…”

Section: Introductionmentioning

confidence: 89%

A weak Kellogg property for quasiminimizers

Björn¹

2006

Comment. Math. Helv.

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Abstract. The Kellogg property says that the set of irregular boundary points has capacity zero, i.e. given a bounded open set there is a set E ⊂ ∂ with capacity zero such that for all p-harmonic functions u in with continuous boundary values in Sobolev sense, u attains its boundary values at all boundary points in ∂ \ E.In this paper, we show a weak Kellogg property for quasiminimizers: a quasiminimizer with continuous boundary values in Sobolev sense takes its boundary values at quasievery boundary point. The exceptional set may however depend on the quasiminimizer. To obtain this result we use the potential theory of quasisuperminimizers and prove a weak Kellogg property for quasisuperminimizers. This is done in complete doubling metric spaces supporting a Poincaré inequality. Mathematics Subject Classification (2000). Primary 31C45; Secondary 35J65, 49J27, 49N60.

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“…For instance, in [12], [20] quasiconformal mappings in metric spaces are studied. Also some results of Euclidean potential theory can be generalized to metric spaces, see [17], [18], [19] and [25]. Thanks to Cheeger's definition of partial derivatives [7], it is even possible to study partial differential equations on such spaces, see [4] and [6].…”

Section: 2) µ(B(y R)) µ(B(x R)) ≥ C R Rmentioning

confidence: 99%