2007 Fiscal Year Final Research Report Summary
On the study of the theory of viscosity solutions and its new developments
| Project/Area Number |
16340032
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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 |
Basic analysis
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| Research Institution | Saitama University |
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Principal Investigator |
KOIKE Shigeaki Saitama University, Fuculty of Science and Engeneering, Professor (90205295)
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| Co-Investigator(Kenkyū-buntansha) |
MORIMOTO Hiroaki Ehime University, Faculty of Science and Engeneering, Professor (80166438)
ISHII Hitoshi Waseda University, Facaulty of Education and Integrated Arts and Sciences, Professor (70102887)
NAGAI Hideo Osaka University, Graduate School of Engeneering Sciences, Professor (70110848)
MIKAMI Toshio Hiroshima University, Faculty of Engeneering, Professor (70229657)
ISHII Katsuynki Kobe University, Fuculty of Maritime Sciences, Associate Professor (40232227)
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| Project Period (FY) |
2004 – 2007
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| Keywords | viscosity solution / fully nonlinear equation / degenerate elliptic equation / uniformly elliptic equation / calculus of variation / Harnack inequality / Mathematical Finance / optimal control |
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| Research Abstract |
The Aleksandrov-Bakelman-Pucci maximum principle for Lp-viscosity solutions of fully nonlinear second order uniformly elliptic/parabolic partial differential equations with possibly superllinear growth terms of the first derivatives, unbounded coefficients, unbounded inhomogeneous terms has been established under appropriate hypotheses in two research papers with A. Swiech. Some counter-examples have been presented when there are no hypotheses.
Perron's method has been first applied to Lp-viscosity solutions of fully nonlinear elliptic partial differential equations by introducing semicontinuous Lp-visosity solutions.
For several nonlinear variational inequalities arising in Mathematical Finance, optimal controls have been constructed by showing that associated value functions admit enough regularity in research papers with H. Morimoto, and H. Morimoto and S. Sakaguchi.
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