| 1. |
- Aikawa, Hiroaki, et al.
(författare)
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Dichotomy of global capacity density in metric measure spaces
- 2018
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Ingår i: Advances in Calculus of Variations. - : WALTER DE GRUYTER GMBH. - 1864-8258 .- 1864-8266. ; 11:4, s. 387-404
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Tidskriftsartikel (refereegranskat)abstract
- The variational capacity cap(p) in Euclidean spaces is known to enjoy the density dichotomy at large scales, namely that for every E subset of R-n, infx is an element of R(n)cap(p)(E boolean AND B(x, r), B(x, 2r))/cap(p)(B(x, r), B(x, 2r)) is either zero or tends to 1 as r -amp;gt; infinity. We prove that this property still holds in unbounded complete geodesic metric spaces equipped with a doubling measure supporting a p-Poincare inequality, but that it can fail in nongeodesic metric spaces and also for the Sobolev capacity in R-n. It turns out that the shape of balls impacts the validity of the density dichotomy. Even in more general metric spaces, we construct families of sets, such as John domains, for which the density dichotomy holds. Our arguments include an exact formula for the variational capacity of superlevel sets for capacitary potentials and a quantitative approximation from inside of the variational capacity.
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| 2. |
- 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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| 3. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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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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| 4. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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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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| 5. |
- Aikawa, Hiroaki, 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)
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| 6. |
- 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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| 7. |
- Aikawa, Hiroaki, 1956, et al.
(författare)
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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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| 8. |
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