The Dirichlet problem for $p$-harmonic functions with respect to arbitrary compactifications

Journal of Mathematical Analysis and Applications

2019

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We use sphericalization to study the Dirichlet problem, Perron solutions and boundary regularity for p-harmonic functions on unbounded sets in Ahlfors regular metric spaces. Boundary regularity for the point at infinity is given special attention. In particular, we allow for several "approach directions" towards infinity and take into account the massiveness of their complements.In 2005, Llorente-Manfredi-Wu showed that the p-harmonic measure on the upper half space R n + , n ≥ 2, is not subadditive on null sets when p = 2. Using their result and spherical inversion, we create similar bounded examples in the unit ball B ⊂ R n showing that the n-harmonic measure is not subadditive on null sets when n ≥ 3, and neither are the p-harmonic measures in B generated by certain weights depending on p = 2 and n ≥ 2.

Section: P-harmonic Functions On Unbounded Domainssupporting

confidence: 72%

Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular spaces

Journal of Mathematical Analysis and Applications

2019

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“…In the proof below we use that it follows from the metrizability that C(∂ 1 Ω) is separable, and instead we could have used this assumption. However, by Theorem 2.10 in Björn-Björn-Sjödin [17] these two assumptions are equivalent. The name "weak Kellogg property" was coined in Björn [2] where it was obtained for quasiminimizers with respect to the given metric boundary.…”

Section: The Kellogg Property and Uniqueness Resultsmentioning

confidence: 97%

“…As ∂ 1 Ω is Sobolev-resolutive, it is automatically resolutive. Moreover it is q.e.-invariant by Proposition 7.1 in Björn-Björn-Sjödin[17]. It follows from Theorem 7.1 that the Kellogg property holds.…”

mentioning

confidence: 76%

“…To see this one just need to take restrictions to Ω ′ 2 , and apply the same resolutivity results. Note however that it is not trivial that the restriction of a C p ( · ; Ω 2 )quasicontinuous function is C p ( · ; (Ω ′ ) 2 )-quasicontinuous, since the conditions (3.1) and (3.2) are different, but this follows from the fact that C p ( · ; Ω 2 ) is an outer capacity, by Proposition 4.2 in [17].…”

Section: Regularity As a Local Propertymentioning

confidence: 99%

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Correction of ‘The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications’

Complex Variables and Elliptic Equations

2019

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In this paper boundary regularity for p-harmonic functions is studied with respect to the Mazurkiewicz boundary and other compactifications. In particular, the Kellogg property (which says that the set of irregular boundary points has capacity zero) is obtained for a large class of compactifications, but also two examples when it fails are given. This study is done for complete metric spaces equipped with doubling measures supporting a p-Poincaré inequality, but the results are new also in unweighted Euclidean spaces.Note that by Proposition 3.5, C p (∂ M Ω, Ω M ) > 0, so the Kellogg property is never trivial. The following uniqueness result is also new. Theorem 1.3. Assume that Ω is a bounded domain which is finitely connected at the boundary. Let f ∈ C(∂ M Ω). Then there exists a unique bounded p-harmonic function u on Ω such thatMoreover, u equals the Perron solution P Ω M f .We are also able to show that boundary regularity is a local property for the Mazurkiewicz boundary in the following sense.Theorem 1.4. Assume that Ω is a bounded domain which is finitely connected at the boundary. Letx ∈ ∂ M Ω and let G be an Ω M -neighbourhood ofx.Thenx is regular with respect to Ω M if and only if it is regular with respect toThroughout the paper we also study to what extent such results are true for other compactifications of Ω. The details describing the different results and cases are quite involved.Boundary regularity for p-harmonic functions with respect to the given metric boundary has been studied for a long period, especially on R n . The first significant result was Maz ′ ya's [37] sufficiency part of the Wiener criterion in 1970. Later on the full Wiener criterion was obtained in various situations including weighted R n and for Cheeger p-harmonic functions on metric spaces, see [25], [32], [35], [39] and [18]. The full Wiener criterion remains open (for the given metric boundary) in the generality considered here, but the sufficiency has been obtained, see [21] and [19], and a weaker necessity condition, see [20].Abstract. We fill in a gap in the proofs of Theorems 1.1-1.4 in "The Kellogg property and boundary regularity for p-harmonic functions with respect to the Mazurkiewicz boundary and other compactifications", to appear in

“…As already mentioned, the Perron method for p-harmonic functions was extended to metric spaces in Björn-Björn-Shanmugalingam [17] and Hansevi [30]. It has also been extended to other types of boundaries in [19], [20], [27], and [7]. Various aspects of boundary regularity for p-harmonic functions on bounded open sets in metric spaces have also been studied in [2], [4]- [10] and [13].…”

Section: Introductionmentioning

confidence: 99%

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

Analysis and Geometry in Metric Spaces

Hansevi

2019

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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.