• Search Research Projects
  • Search Researchers
  • How to Use
  1. Back to previous page

Construction of analysis via partitions and weights of spaces

Research Project

Project/Area Number 21K18587
Research Category

Grant-in-Aid for Challenging Research (Exploratory)

Allocation TypeMulti-year Fund
Review Section Medium-sized Section 12:Analysis, applied mathematics, and related fields
Research InstitutionKyoto University

Principal Investigator

KIGAMI JUN  京都大学, 情報学研究科, 教授 (90202035)

Project Period (FY) 2021-07-09 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥6,370,000 (Direct Cost: ¥4,900,000、Indirect Cost: ¥1,470,000)
Fiscal Year 2023: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2022: ¥2,080,000 (Direct Cost: ¥1,600,000、Indirect Cost: ¥480,000)
Fiscal Year 2021: ¥2,210,000 (Direct Cost: ¥1,700,000、Indirect Cost: ¥510,000)
Keywords距離空間 / 空間の分割 / フラクタル / グラフ近似 / ソボレフ空間 / 解析学 / 分割 / 重み
Outline of Research at the Start

フラクタルに代表される滑らかさのない空間は、自然界をモデル化する際にも重要な役割を果たすことが知られている。従って、自然界での物理現象を記述するためには、フラクタルのような複雑な空間上で、ユークリッド空間などの滑らかな空間での「微分方程式論」やそれに付随するソボレフ空間などの関数空間の理論に相当する数学が必要となる。本研究では、複雑な空間上に微分方程式論を展開するための基礎となるソボレフ空間論などの「解析学」を構築することを目指している。

Outline of Final Research Achievements

In this study, we tried to construct a counterpart of (1, p)-Sobolev spaces on metric spaces via a scaling of discrete p-energies on discrete graphs approximating the original metric space. Note that we can not use the notion of derivation on complex metric spaces like fractals. In conclusion, we establish the notion of p-conductive homogeneity under which we have a proper scaling constant of discrete p-energies and as a consequence, we obtain a counterpart of (1, p)-Sobolev spaces and p-energies. In particular, in the case p = 2, we succeed to construct a natural diffusion process on the metric space.

Academic Significance and Societal Importance of the Research Achievements

この研究で考察の対象となる空間はフラクタルなどの複雑な空間である。フラクタルはマンデルブローにより自然界の物体の適切なモデルとして提唱された。従来のユークリッド空間や可微分多様などの滑らかな空間をモデルとした物理学ではその上の物理現象は微分を用いて記述されるが、フラクタルなどの複雑な形状を持つ空間では微分を定義することが出来ない。すなわち、フラクタルをモデルとした物体上の物理現象を記述するためには、「微分を用いない解析学」の構築が必要となる。本研究はそのような微分が定義できないような複雑な空間上での解析学の基礎の確立を目指している。

Report

(4 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • Research Products

    (8 results)

All 2024 2023 2022 Other

All Int'l Joint Research (1 results) Journal Article (2 results) (of which Peer Reviewed: 2 results,  Open Access: 2 results) Presentation (4 results) (of which Int'l Joint Research: 4 results,  Invited: 4 results) Book (1 results)

  • [Int'l Joint Research] University of British Columbia(カナダ)

    • Related Report
      2021 Research-status Report
  • [Journal Article] “The Sierpinski gasket minus its bottom line” as a tree of Sierpinski gaskets2024

    • Author(s)
      J. Kigami & K. Takahashi
    • Journal Title

      Math. Z.

      Volume: 28 Issue: 2

    • DOI

      10.1007/s00209-023-03416-1

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Conductive homogeneity of compact metric spacdes and condtruction of p-energy2023

    • Author(s)
      Jun KIGAMI
    • Journal Title

      Memoirs of the European Mathematical Society

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Conductive homogeneity of locally symmetric polygon-based self-similar sets2024

    • Author(s)
      J. Kigami
    • Organizer
      CIRM workshop "Analysis on fractals and networks, and applications"
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Construction of Sobolev spaces on metric spaces2024

    • Author(s)
      J. Kigami
    • Organizer
      Fractals, quantum graphs and applications in pure and applied sciences
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Yet another construction of "Sobolev spaces" on metric spaces2023

    • Author(s)
      J. Kigami
    • Organizer
      Quasiworld workshop
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Conductive homogeneity of compace metric spaces and construction of p-energy2022

    • Author(s)
      Jun Kigami
    • Organizer
      Analysis on Metric spaces
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research / Invited
  • [Book] Conductive homogeneity of compact metric spaces and construction of p-energy2023

    • Author(s)
      J. Kigami
    • Total Pages
      129
    • Publisher
      European Math. Society
    • ISBN
      9783985470563
    • Related Report
      2023 Annual Research Report

URL: 

Published: 2021-07-13   Modified: 2025-01-30  

Information User Guide FAQ News Terms of Use Attribution of KAKENHI

Powered by NII kakenhi