| 1. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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MARTIN BOUNDARY FOR UNION OF CONVEX SETS
- 2002
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Ingår i: 京都大学数理解析研究所, Potential Theory and Related Topics. ; 1293, s. 1-14
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Tidskriftsartikel (refereegranskat)abstract
- We study Martin boundary points of aproper subdomain in $\mathbb{R}^{n}$ , where $n$ $\geq 2$ , that can be represented as the union of open convex sets. Especially, we give acertain sufficient condition for aboundary point to have exactly one (minimal) Martin boundary point.
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| 2. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
-
Martin boundary of a fractal domain
- 2003
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Ingår i: Potential Analysis. - : Springer Science and Business Media LLC. - 0926-2601 .- 1572-929X. ; 18:4, s. 311-357
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Tidskriftsartikel (refereegranskat)abstract
- A uniformly John domain is a domain intermediate between a John domain and a uniform domain. We determine the Martin boundary of a uniformly John domain D as an application of a boundary Harnack principle. We show that a certain self-similar fractal has its complement as a uniformly John domain. In particular, the complement of the 3-dimensional Sierpinacuteski gasket is a uniform domain and its Martin boundary is homeomorphic to the Sierpinacuteski gasket itself.
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| 3. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
-
Martin boundary points of a John domain and unions of convex sets
- 2006
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Ingår i: J. Math. Soc. Japan. - 0025-5645 .- 1881-1167. ; 58:1, s. 247-274
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Tidskriftsartikel (refereegranskat)abstract
- We show that a John domain has finitely many minimal Martin boundary points at each Euclidean boundary point. The number of minimal Martin boundary points is estimated in terms of the John constant. In particular, if the John constant is bigger than $\sqrt3/2$ , then there are at most two minimal Martin boundary points at each Euclidean boundary point. For a class of John domains represented as the union of convex sets we give a sufficient condition for the Martin boundary and the Euclidean boundary to coincide.
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| 4. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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On boundary layers
- 1996
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Ingår i: Arkiv för matematik. ; 34:1
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Tidskriftsartikel (refereegranskat)abstract
- https://projecteuclid.org/download/pdf_1/euclid.afm/1485898494
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| 5. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
-
The 3G inequality for a uniformly John domain
- 2005
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Ingår i: Kodai Mathematical Journal. ; 28:2, s. 209-219
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Tidskriftsartikel (refereegranskat)abstract
- Let G be the Green function for a domain D $\subset$ Rd with d ≥ 3. The Martin boundary of D and the 3G inequality: $\frac{G(x,y)G(y,z)}{G(x,z)} \le A(|x-y|^{2-d}+|y-z|^{2-d})$ for x,y,z $\in$ D are studied. We give the 3G inequality for a bounded uniformly John domain D, although the Martin boundary of D need not coincide with the Euclidean boundary. On the other hand, we construct a bounded domain such that the Martin boundary coincides with the Euclidean boundary and yet the 3G inequality does not hold.
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