Boundary Behaviour of ${\scr M}$ -Harmonic Functions and Non-Isotropic Hausdorff Measure

Let D be a domain in Euclidean space of d dimensions and K a compact subset of D. The well known Harnack inequality assures the existence of a positive constant A depending only on D and K such that (l/A)u(x)<u(y)<Au(x) for all x and y in K and all positive harmonic functions u on D. In this we obtain a global integral version of this inequality under geometrical conditions on the domain. The result is the following: suppose D is a Lipschitz domain satisfying the uniform exterior sphere condition—stated in Section 2. If u is harmonic in D with continuous boundary data f thenwhere ds is the d — 1 dimensional Hausdorff measure on the boundary ժD. A large class of domains satisfy this condition. Examples are C2-domains, convex domains, etc.

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Boundary behaviour of p-harmonic functions in domains beyond Lipschitz domains

Advances in Calculus of Variations ◽

10.1515/acv.2008.005 ◽

2008 ◽

Vol 1 (2) ◽

Cited By ~ 16

Author(s):

John L. Lewis ◽

Kaj Nyström

Keyword(s):

Harmonic Functions ◽

Lipschitz Domains ◽

Boundary Behaviour

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Boundary behaviour for p harmonic functions in Lipschitz and starlike Lipschitz ring domains

Annales Scientifiques de l École Normale Supérieure ◽

10.1016/j.ansens.2007.09.001 ◽

2007 ◽

Vol 40 (5) ◽

pp. 765-813 ◽

Cited By ~ 25

Author(s):

J LEWIS ◽

K NYSTROM

Keyword(s):

Harmonic Functions ◽

Boundary Behaviour ◽

Ring Domains

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Hausdorff measure and extension results for subharmonic functions, for separately subharmonic functions, for harmonic functions and for separately harmonic functions

Subharmonic Functions, Generalizations, Holomorphic Functions, Meromorphic Functions, and Properties ◽

10.2174/9789811498701121010010 ◽

2021 ◽

pp. 92-104

Author(s):

Juhani Riihentaus

Keyword(s):

Hausdorff Measure ◽

Harmonic Functions ◽

Subharmonic Functions

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$$p$$ -Harmonic functions in the Heisenberg group: boundary behaviour in domains well-approximated by non-characteristic hyperplanes

Mathematische Annalen ◽

10.1007/s00208-013-0896-3 ◽

2013 ◽

Vol 357 (1) ◽

pp. 307-353 ◽

Cited By ~ 2

Author(s):

Kaj Nyström

Keyword(s):

Heisenberg Group ◽

Harmonic Functions ◽

Boundary Behaviour ◽

Group Boundary

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A boundary estimate for harmonic functions

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100059132 ◽

1982 ◽

Vol 91 (1) ◽

pp. 79-90 ◽

Cited By ~ 3

Author(s):

P. J. Rippon

Keyword(s):

Euclidean Space ◽

Harmonic Functions ◽

Conformal Mappings ◽

Dimensional Euclidean Space ◽

Boundary Behaviour ◽

Boundary Estimate ◽

Distortion Theorems

In this paper we extend to certain domains in m-dimensional Euclidean space Rm, m ≥ 3, some results about the boundary behaviour of harmonic functions which, in R2, are known to follow from distortion theorems for conformal mappings.

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