| 研究課題/領域番号 |
22K13947
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| 研究機関 | 沖縄科学技術大学院大学 |
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研究代表者 |
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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| 研究期間 (年度) |
2022-04-01 – 2025-03-31
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| キーワード | Absolute Continuity / Sobolev mappings / Metric measure spaces / Quasiconformal mappings / Lusin property / Nonlocal functional |
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| 研究実績の概要 |
In the first project, we consider Q-absolutely continuous mappings between a doubling metric measure space and a Banach space. The relation between these mappings and Sobolev mappings in supercritical cases is investigated. In particular, we show that pseudomonotone mappings satisfying a relaxed quasiconformality condition are also Q-absolutely continuous.
In the second project, we study a characterization of BV and Sobolev functions via nonlocal functionals in metric spaces equipped with a doubling measure and supporting a Poincare inequality. Compared with previous works, we consider more general functionals. We also give a counterexample in the case p=1 demonstrating that in metric measure spaces the limit of the nonlocal functions is only comparable to the variation measure.
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| 現在までの達成度 (区分) |
現在までの達成度 (区分)
2: おおむね順調に進展している
理由
The first project studies absolutely continuous mappings and Lusin property of quasiconformal mappings and yields new findings. The result has been accepted for publication at Manuscripta Mathematica.
The second project focuses on characterization of functions of bounded variation and Sobolev functions on metric measure spaces using nonlocal functionals. The results we obtained extend the previous work to a more general class of nonlocal functionals. The paper has been submitted.
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| 今後の研究の推進方策 |
We propose the following directions for our work in the next step.
1. Focus on one of the main question raised in our proposal that investigating Sobolev regularity of quasiconformal mappings with relaxed space and homeomorphism conditions
2. Extend the characterization of functions of bounded variation and Sobolev functions on metric measure spaces using nonlocal functionals to other functional classes
3. As Green function for p-Laplacian has been applied widely in the study of quasiconformal mappings, we also propose to study the Green function of p-Laplace equation in metric measure spaces.
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| 次年度使用額が生じた理由 |
I will use the budget for research travel, hosting visitors at OIST and purchasing reference books.
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