Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces

Björn, Anders; Björn, Jana

doi:10.2969/jmsj/1179759546

Cited by 43 publications

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References 25 publications

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“…A standing assumption in [6] was that is a nonempty bounded open set with C p (X \ ) > 0. However, the proof of this lemma in [6] does not use the boundedness nor the assumption C p (X \ ) > 0 (the lemma is trivial in the case when C p (X \ ) = 0) and thus the lemma holds as stated here.…”

Section: Lemma 35 Assume That C P (X \ ) > 0 and Let F ∈ C(∂ ) ∩ N mentioning

confidence: 99%

“…for p-harmonic functions) can be defined as in the introduction, or equivalently in a manner similar to Definition 5.1, see Björn-Björn [6], Theorem 6.1, where several characterizations of regular boundary points can be found. It is immediate that being regular for quasiharmonic functions is a stronger requirement.…”

Section: Boundary Regularitymentioning

confidence: 99%

“…(A p-harmonic function is a continuous minimizer and a quasiharmonic function is a continuous quasiminimizer. For the relation between Sobolev and Perron solutions of the Dirichlet problem, see Björn-Björn-Shanmugalingam [8] and Björn-Björn [6]. )…”

Section: Introductionmentioning

confidence: 99%

“…1-quasisuperharmonic functions) is equivalent to regularity for p-harmonic functions, see Björn-Björn [6], Theorem 6.1. We do not know if regularity for quasiharmonic functions is equivalent to regularity for quasisuperharmonic functions, but it is clear that the latter implies the former.…”

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confidence: 99%

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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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Section: Lemma 35 Assume That C P (X \ ) > 0 and Let F ∈ C(∂ ) ∩ N mentioning

confidence: 99%

Section: Boundary Regularitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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A weak Kellogg property for quasiminimizers

Björn¹

2006

Comment. Math. Helv.

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“…The obstacle problem has been studied for bounded sets in (weighted) R n (see, e.g., Heinonen-Kilpeläinen-Martio [20] and the references therein), and later also for bounded sets in more general metric spaces (see, e.g., Björn-Björn [3], [4], [5], Björn-Björn-Mäkäläinen-Parviainen [6], Björn-Björn-Shanmugalingam [8], Kinnunen-Martio [25], Kinnunen-Shanmugalingam [26], and Shanmugalingam [31]). The double obstacle problem has also been studied (see, e.g., Farnana [13,14,15,16] and Eleuteri-Farnana-Kansanen-Korte [12]).…”

Section: Introductionmentioning

confidence: 99%

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

Hansevi¹

2015

Ann. Acad. Sci. Fenn. Math.

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Abstract. The obstacle problem associated with p-harmonic functions is extended to unbounded open sets, whose complement has positive capacity, in the setting of a proper metric measure space supporting a (p, p)-Poincaré inequality, 1 < p < ∞, and the existence of a unique solution is proved. Furthermore, if the measure is doubling, then it is shown that a continuous obstacle implies that the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.

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Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers

Björn

Мартио

2009

Results. Math.

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Quasiharmonic functions correspond to p-harmonic functions when minimizers of the p-Dirichlet integral are replaced by quasiminimizers. In this paper, boundary regularity for quasiminimizers is characterized in several ways; in particular it is shown that regularity is a local property of the boundary. For these characterizations we employ a version of the so called pasting lemma; this is a useful tool in the theory of superharmonic functions and our version extends the classical pasting lemma to quasisuperharmonic functions and quasisuperminimizers.The results are obtained for metric measure spaces, but they are new also in the Euclidean spaces. Mathematics Subject Classification (2000). Primary: 31C45; Secondary: 31B25, 35J60, 35J65.

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Boundary regularity for p-harmonic functions and solutions of the obstacle problem on metric spaces

Cited by 43 publications

References 25 publications

A weak Kellogg property for quasiminimizers

A weak Kellogg property for quasiminimizers

The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in R^n and metric spaces

Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers

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