2022 Fiscal Year Research-status Report
Quasiconformal and Sobolev mappings on metric measure spaces
| Project/Area Number |
22K13947
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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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| Keywords | Absolute Continuity / Sobolev mappings / Metric measure spaces / Quasiconformal mappings / Lusin property / Nonlocal functional |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
I will use the budget for research travel, hosting visitors at OIST and purchasing reference books.
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