| Project/Area Number |
09440062
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| Research Category |
Grant-in-Aid for Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
解析学
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| Research Institution | Shimane University |
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Principal Investigator |
AIKAWA H. Shimane Univ., Dept.of Math.Professor, 総合理工学部, 教授 (20137889)
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| Co-Investigator(Kenkyū-buntansha) |
MURATA M. Tokyo Inst.Tech.Dept.of Math.Professor, 理学部, 教授 (50087079)
MIZUTA M. Hiroshima Univ., Dept.of Math.Professor, 総合科学部, 教授 (00093815)
SUGIE J. Shimane Univ., Dept.of Math.Professor, 総合理工学部, 教授 (40196720)
FURUMOCHI T. Shimane Univ., Dept.of Math.Professor, 総合理工学部, 教授 (40039128)
YAMASAKI M. Shimane Univ., Dept.of Math.Professor, 総合理工学部, 教授 (70032935)
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| Project Period (FY) |
1997 – 1998
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| Keywords | potential / harmonic / superharmonic / integrability / coarea fomula / capacity / fine topology / extremal length |
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| Research Abstract |
The research results of the main investigator are mainly stated.
Precise properties of solutions to an elliptic differential equation is given by the Green function and the Green operator, We have studied the following :
(i) Estimate of the norm of the Green operator.
(ii) Green function under perturbation.
(iii) Integrability of positive superharmonic functions by perturbation.
(iv) Study of perturbation by coarea formula.
The well-known notion of extremal length has been generalized. The reciprocal relationship between the extremal distance connecting two sets and the extremal width separating them has been established. Moreover, extremal length is extended to vector measures and the relationship to capacity with respect to a degenerate elliptic equation is given. This is a joint work with Prof. M.Ohtsuka.
The generalized Cranston-McConnell inequality for a discontinuous superharmonic function is proved with the aid [of fine topology. This is an extension of the result of the investigator in J.Analyse Math. (1996).
Fine topology, cluster sets, boundary values, boundary Harnack principle and the Martin boundary have been studied with M.Mizutani and S.Gardiner. Papers about them are submitted.
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