| Project/Area Number |
22K13947
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| Research Category |
Grant-in-Aid for Early-Career Scientists
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| Allocation Type | Multi-year Fund |
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| Review Section |
Basic Section 12020:Mathematical analysis-related
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| Research Institution | Okinawa Institute of Science and Technology Graduate University |
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Principal Investigator |
Zhou Xiaodan 沖縄科学技術大学院大学, 距離空間上の解析ユニット, 准教授 (10871494)
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| Project Period (FY) |
2022-04-01 – 2025-03-31
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| Project Status |
Completed (Fiscal Year 2024)
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| Budget Amount *help |
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
Fiscal Year 2024: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2023: ¥1,820,000 (Direct Cost: ¥1,400,000、Indirect Cost: ¥420,000)
Fiscal Year 2022: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | Quasiconformal mappings / weighted spaces / Sobolev functions / Lusin property / BV functions / nonlocal functional / non-local functional / PI spaces / Green function / p-Laplace / Quasiconformal mapping / Newton-Sobolev mapping / metric measure spaces / Poincare inequality / Absolute Continuity / Sobolev mappings / Metric measure spaces / Nonlocal functional / Sobolev mapping |
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| Outline of Research at the Start |
We propose to study the reduced assumptions on the spaces and homeomorphism to ensure the Sobolev regularity and related applications of the regularity results. We aim to develop new techniques systematically for achieving analytic properties and study the applications on metric measure spaces.
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| Outline of Final Research Achievements |
We developed new techniques and obtained the Sobolev regularity of metric quasiconformal mappings under reduced assumptions on the metric measure spaces and homeomorphism. We further applied the results to study the preservation of sets of measure zero of the mappings. We also obtained new characterizations of functions of bounded variation and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincare inequality. In a side project, we study a PDE-based approach to the horizontally quasiconvex envelope of a given continuous function in the Heisenberg group.
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| Academic Significance and Societal Importance of the Research Achievements |
Our main results complement the theory of Sobolev regularity of quasiconformal mappings on a large class of weighted spaces and a wider class of mappings. The results can bring new insights to related existing questions and applications can be expected in physics or nonlinear elasticity.
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