| 研究課題/領域番号 |
22K13947
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| 研究種目 |
若手研究
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| 配分区分 | 基金 |
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| 審査区分 |
小区分12020:数理解析学関連
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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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| 研究課題ステータス |
交付 (2023年度)
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| 配分額 *注記 |
4,550千円 (直接経費: 3,500千円、間接経費: 1,050千円)
2024年度: 1,170千円 (直接経費: 900千円、間接経費: 270千円)
2023年度: 1,820千円 (直接経費: 1,400千円、間接経費: 420千円)
2022年度: 1,560千円 (直接経費: 1,200千円、間接経費: 360千円)
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| キーワード | Quasiconformal mapping / Newton-Sobolev mapping / metric measure spaces / nonlocal functional / Poincare inequality / Absolute Continuity / Sobolev mappings / Metric measure spaces / Quasiconformal mappings / Lusin property / Nonlocal functional / Sobolev mapping |
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| 研究開始時の研究の概要 |
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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| 研究実績の概要 |
The first project concerns the Sobolev regularity of mappings satisfying the metric condition of quasiconformality outside suitable exceptional sets. Contrary to previous works, we only assume an asymptotic version of Ahlfors-regularity on the spaces. Already in the classical setting, our theory detects Sobolev mappings that are not recognized by previous results.
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.
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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
One of the main questions raised in our proposal that investigating Sobolev regularity of quasiconformal mappings with relaxed space and homeomorphism conditions has been answered in the first project. The results appear in two published paper this year.
The second project on the characterization of BV and Sobolev via nonlocal functional has been published as well.
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| 今後の研究の推進方策 |
The next step mainly consists of two projects:
1. Study the asymptotic behavior of three classes of nonlocal functionals in complete metric spaces equipped with a doubling measure and supporting a Poincare inequality.
2. Study the Green function of Q-Laplace equation in Q-regular metric measure spaces as this function class has been applied widely in the study of quasiconformal mappings.
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