Pasting Lemmas and Characterizations of Boundary Regularity for Quasiminimizers

Björn, Anders; Мартио, Олли

doi:10.1007/s00025-009-0437-2

Cited by 9 publications

(13 citation statements)

References 17 publications

(34 reference statements)

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“…Note that semiregularity and semiregularity for quasiminimizers coincide, by Björn [8]. For more on boundary regularity for quasiminimizers see Ziemer [31], J. Björn [18], Martio [28], Björn [5,8,9], Björn-Björn [10] and Björn-Martio [17].…”

Section: Theorem 26 (The Wiener Criterion) the Point X 0 ∈ ∂ω Is Rementioning

confidence: 98%

p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

Björn

2010

Journal of Differential Equations

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For p-harmonic functions on unweighted R 2 , with 1 < p < ∞, we show that if the boundary values f has a jump at an (asymptotic) corner point z 0 , then the Perron solution P f is asymptotically a + b arg(z − z 0 ) +o(|z − z 0 |). We use this to obtain a positive answer to Baernstein's problem on the equality of the p-harmonic measure of a union G of open arcs on the boundary of the unit disc, and the pharmonic measure of G. We also obtain various invariance results for functions with jumps and perturbations on small sets. For p > 2 these results are new also for continuous functions. Finally we look at generalizations to R n and metric spaces.

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Section: Theorem 26 (The Wiener Criterion) the Point X 0 ∈ ∂ω Is Rementioning

confidence: 98%

p-Harmonic functions with boundary data having jump discontinuities and Baernstein's problem

Björn

2010

Journal of Differential Equations

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“…Regularity for quasiharmonic functions was called regularity for quasiminimizers in Björn-Martio [15]. In Björn [6], it was shown that if u and f are as above and x 0 is strongly irregular for quasiharmonic functions, then, in our terms, f (x 0 ) ∈ C(u, x 0 , ).…”

Section: Boundary Regularity For Quasiharmonic Functionsmentioning

confidence: 96%

“…In Björn-Martio [15], the following characterization was given. It was also observed there that it holds if "quasi" is replaced by "Q-quasi" throughout.…”

Section: Theorem 72 (Weak Kellogg Property For Quasiharmonic Functiomentioning

confidence: 98%

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Cluster sets for sobolev functions and quasiminimizers

Björn

2010

JAMA

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In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set has boundary values f in Sobolev sense and f | ∂ is continuous at x 0 ∈ ∂ , then the essential cluster set C(u, x 0 , ) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.

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“…Most aspects of the higher-dimensional theory fit just as well in metric spaces, and this theory, in particular concerning boundary regularity, has recently been developed further in a series of papers by Martio [17]- [19], A. Björn-Martio [7], A. Björn [1]- [4] and J. Björn [9].…”

Section: Introductionmentioning

confidence: 99%