2019 Fiscal Year Final Research Report
Generalization of Nearby cycles for D-modules and its application to boundary value problem
| Project/Area Number |
15K04908
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Basic analysis
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| Research Institution | Nihon University |
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Principal Investigator |
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| Project Period (FY) |
2015-04-01 – 2020-03-31
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| Keywords | D加群 / 境界値問題 / 超局所解析 / 佐藤超函数 / 近接輪体 |
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| Outline of Final Research Achievements |
In the framework of algebraic analysis, we extend definitions of nearby and vanishing cycle modules to Fuchsian D-modules in the sense of Laurent-Monteiro Fernandes, and obtain unique solvability results in the complex domain in holomorphic category. We also give distinguished triangles connecting nearby and vanishing cycle modules, ordinary and extraordinary inverse images, that are related by the octahedral axiom. These distinguished triangles induce the relations between solution sheaves of holomorpic functions, of specializations of holomorpic functions, of holmorphic hyperfunctions, and of holomorphic microfunctions.
As an application, a general boundary value morphism is defined for any hyperfunction solutions to the Fuchsian D-modules in derived category, and the injectivity of this morphism in zero-th cohomology is proved (uniqueness of the solution). Moreover, under a kind of hyperbolicity condition, it is proved that this morphism is surjective (solvability).
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| Free Research Field |
数学
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| Academic Significance and Societal Importance of the Research Achievements |
或る種の形式的無限階偏微分作用素を導入する事で，複素領域に於いて Fuchs 型 D 加群に対しても，正則特殊化可能 D 加群と同様の結果を証明する事が出来た．結果として，D 加群論に貢献出来たと考えている．
更にその応用として，実領域に於いて，非特性型，特性型の両方を含む Fuchs 型 D 加群の佐藤超函数解に対し，線型偏微分方程式論の最も基本的な問題の一つである境界値問題の定式化が成功した事は重要であると考える．特に解の一意性，及び或る種の双曲性条件の元での可解性等，満足すべき結果が得られた．
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