1992 Fiscal Year Final Research Report Summary
Potential-theoretic study on the behavior of solutions of partial differential equations
| Project/Area Number |
03452010
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| Research Category |
Grant-in-Aid for General Scientific Research (B)
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| Allocation Type | Single-year Grants |
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| Research Field |
解析学
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| Research Institution | HIROSHIMA UNIVERSITY |
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Principal Investigator |
MAEDA Fumiyuki Hiroshima U., Fac. of Sci., Prof., 理学部, 教授 (10033804)
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| Co-Investigator(Kenkyū-buntansha) |
SHIBA Masakazu Hiroshima U., Fac. of Sci., Asso. Prof., 理学部, 助教授 (70025469)
OHARU Shinnosuke Hiroshima U., Fac. of Sci., Prof., 理学部, 教授 (40063721)
NAITO Manabu Hiroshima U., Fac. of Sci., Asso. Prof., 理学部, 助教授 (00106791)
KUSANO Takasi Hiroshima U., Fac. of Sci., Prof., 理学部, 教授 (70033868)
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| Project Period (FY) |
1991 – 1992
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| Keywords | Potential theory / Partial differential equation / Elliptic equation / Parabolic equation / Riemann surface / Harmonic space / Balayage space / Dual structure |
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| Research Abstract |
Concerning the global behavior of solutions of nonlinear equations, Kusano, Naito, Kura and Oharu have obtained the following results: for entire solutions of certain second and higher order semilinear or quasilinear elliptic equations, the structure of positive solutions and the oscillation theory such as the number of zeros of spherically symmetric solutions can be clarified by using the results and methods for similar ordinary differential equations; the supersolution-subsolution method can be applied to higher order elliptic equations; the existence of positive solutions with various asymptotic behaviors and their construction method for the p-Laplace equation in an exterior domain, the oscillation properties of solutions of parabolic functional partial differential equations with oscillating coefficients. On the other hand, Shiba and Masumoto investigated the continuation problem of Riemann surfaces; potential theoretic realization problems as well as relations with classical function theory. They have given estimates of the span of an open Riemann surface of genus 1, some new results on extrenal problems for the exterior problem by parallel treatment of the cases of genus 0 and 1 and relations between some conformal distances and the hyperbolic distance. The head investigator has shown the following: in the theory of harmonic spaces, which provides a unified potential theoretic treatment of the theory of elliptic and parabolic partial differential equations, if its dual structure is defined in terms of measure representations, we can derive Green's formulae and furthermore, on more general balayage spaces, duality of two structures can be defined in the same way.
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