2023 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 | Quasiconformal mapping / Newton-Sobolev mapping / metric measure spaces / nonlocal functional / Poincare inequality |
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| Outline of Annual Research Achievements |
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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| Current Status of Research Progress |
Current Status of Research Progress
2: Research has progressed on the whole more than it was originally planned.
Reason
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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| Strategy for Future Research Activity |
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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| Causes of Carryover |
The budget will be used for supporting research travels to join international conferences and to visit collaborators, and to host a research workshop at OIST.
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