F-harmonic measure in space

Granlund, Seppo; Lindqvist, Peter; Мартио, Олли

doi:10.5186/aasfm.1982.0717

Cited by 25 publications

(10 citation statements)

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“…The advantage of the Perron method lies in the fact that it allows us to construct a reasonable solution to the Dirichlet problem for boundary data which are not necessarily continuous. The study of Perron solutions has been extended to degenerate elliptic operators in Euclidean domains in [17,28,12], and Granlund-Lindqvist-Martio [10]. Recent development in the study of Perron solutions has been in the direction of applying the method to subelliptic operators, see Markina-Vodop'yanov [31,32].…”

Section: Introductionmentioning

confidence: 99%

The Perron method for p-harmonic functions in metric spaces

Björn

Shanmugalingam

2003

Journal of Differential Equations

134

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We use the Perron method to construct and study solutions of the Dirichlet problem for p-harmonic functions in proper metric measure spaces endowed with a doubling Borel measure supporting a weak (1,q)-Poincaré inequality (for some 1q<p). The upper and lower Perron solutions are constructed for functions defined on the boundary of a bounded domain and it is shown that these solutions are p-harmonic in the domain. It is also shown that Newtonian (Sobolev) functions and continuous functions are resolutive, i.e. that their upper and lower Perron solutions coincide, and that their Perron solutions are invariant under perturbations of the function on a set of capacity zero. We further study the problem of resolutivity and invariance under perturbations for semicontinuous functions. We also characterize removable sets for bounded p-(super)harmonic functions.Original Publication:Anders Björn, Jana Björn and Nageswari Shanmugalingam, The Perron method for p-harmonic functions in metric spaces, 2003, Journal of Differential Equations, (195), 2, 398-429.http://dx.doi.org/10.1016/S0022-0396(03)00188-8Copyright: Elsevier Science B.V., Amsterdamhttp://www.elsevier.com

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Section: Introductionmentioning

confidence: 99%

The Perron method for p-harmonic functions in metric spaces

Björn

Shanmugalingam

2003

Journal of Differential Equations

134

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“…In [9], Granlund, Lindqvist and Martio construct functions which measure the smallness of subsets of the boundary qW in a manner analogous to the 2-harmonic measure. Indeed, their construction yields a p-harmonic function as follows (they only define pharmonic measures for compact sets).…”

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confidence: 99%

The Dirichlet problem for p-harmonic functions on metric spaces

Björn¹,

Björn²,

Shanmugalingam³

2003

Journal Für Die Reine Und Angewandte Mathematik (Crelles Journal)

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We study the Dirichlet problem for p-harmonic functions (and p-energy minimizers) in bounded domains in proper, pathconnected metric measure spaces equipped with a doubling measure and supporting a Poincaré inequality. The Dirichlet problem has previously been solved for Sobolev type boundary data, and we extend this result and solve the problem for all continuous boundary data. We study the regularity of boundary points and prove the Kellogg property, i.e. that the set of irregular boundary points has zero p-capacity.We also construct p-capacitary, p-singular and p-harmonic measures on the boundary. We show that they are all absolutely continuous with respect to the p-capacity. For p ¼ 2 we show that all the boundary measures are comparable and that the singular and harmonic measures coincide. We give an integral representation for the solution to the Dirichlet problem when p ¼ 2, enabling us to extend the solvability of the problem to L 1 boundary data in this case. Moreover, we give a trace result for Newtonian functions when p ¼ 2. Finally, we give an estimate for the Hausdor¤ dimension of the boundary of a bounded domain in Ahlfors Q-regular spaces. Introduction Recent developments in analysis on metric spaces include a study ofSobolev type spaces on metric measure spaces; see Hajłasz [10], Hajłasz-Koskela [11], Heinonen-Koskela [14], Cheeger [6], Franchi-Hajłasz-Koskela [7], Kilpeläinen-Kinnunen-Martio [22], Kinnunen-Latvala [23], Kinnunen-Martio [24], [25], Koskela-MacManus [27], Shanmugalingam [35], and the references therein. A construction of a Sobolev type space, called the Newtonian space, was given in [35] and subsequently studied in Shanmugalingam [36], Kinnunen-Shanmugalingam [26], Bjö rn [3], Bjö rn-MacManus-Shanmugalingam [4], Shanmugalingam [34], Holopainen-Shanmugalingam [17], and Kallunki-Shanmugalingam [21]. The Sobolev type space constructed by Cheeger in [6] yields the same space studied in [35], and enables us to use the deep results of [6]. Moreover the Sobolev type spaces introduced by Hajłasz [10] coincide with our spaces in the case when the metric space satisfies our general assumptions. Brought to you by | University of Connecticut Authenticated Download Date | 5/16/15 11:57 PM Brought to you by | University of Connecticut Authenticated Download Date | 5/16/15 11:57 PMExamples of metric measure spaces that have the above-mentioned properties include the Euclidean spaces (with or without weights), complete manifolds with negatively pinched curvature and positive injectivity radius, Carnot groups such as the Heisenberg group as well as other sub-Riemannian manifolds that arise as Gromov-Hausdor¤ limits of certain Riemannian manifolds, the graphs with uniformly bounded geometry studied by Holopainen and Soardi in [18] and [19], as well as the metric spaces constructed by Bourdon-Pajot [5] and by Laakso [28].Acknowledgement.

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“…This result follows also from arguments in [2]. The purpose of this paper is to show that for dimensions n > 2 there is no lower bound for the order that tends to ∞ as the number of asymptotic values grows to ∞.…”

Section: Introductionmentioning

confidence: 56%

Failure of the Denjoy theorem for quasiregular maps in dimension $n \ge 3$

Holopainen

Rickman

1996

Proc. Amer. Math. Soc.

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Abstract. In 1929 L. V. Ahlfors proved the Denjoy conjecture which states that the order of an entire holomorphic function of the plane must be at least k if the map has at least 2k finite asymptotic values. In this paper, we prove that the Denjoy theorem has no counterpart in the classical form for quasiregular maps in dimensions n ≥ 3. We construct a quasiregular map of R n , n ≥ 3, with a bounded order but with infinitely many asymptotic limits. Our method also gives a new construction for a counterexample of Lindelöf's theorem for quasiregular maps of B n , n ≥ 3.

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F-harmonic measure in space

Cited by 25 publications

References 0 publications

The Perron method for p-harmonic functions in metric spaces

The Perron method for p-harmonic functions in metric spaces

The Dirichlet problem for p-harmonic functions on metric spaces

Failure of the Denjoy theorem for quasiregular maps in dimension $n \ge 3$

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