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Hamilton-Jacobi equations on metric measure spaces

Research Project

Project/Area Number 20K22315
Research Category

Grant-in-Aid for Research Activity Start-up

Allocation TypeMulti-year Fund
Review Section 0201:Algebra, geometry, analysis, applied mathematics,and related fields
Research InstitutionOkinawa Institute of Science and Technology Graduate University

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)

Project Period (FY) 2020-09-11 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥2,860,000 (Direct Cost: ¥2,200,000、Indirect Cost: ¥660,000)
Fiscal Year 2021: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2020: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
KeywordsEikonal equation / viscosity solution / metric spaces / Heisenberg group / differential games / convexity / eikonal equation / metric measure spaces / discontinuous data / h-quasiconvex functions / metric measure space / h-quasiconvexity / Hamilton-Jacobi equation / viscosity solutions / HJ equations / well-posedness
Outline of Research at the Start

Motivated by the rapid developments of optimal transport, control theory, data sciences etc., there is a growing interest in studying the nonlinear PDEs on metric measure spaces. We propose to focus on first order Hamilton-Jacobi equations and investigate the well-posedness on metric spaces.

Outline of Final Research Achievements

First, showing the equivalence of solutions of eikonal equations in metric spaces by introducing a new simple notions called Monges solution. Exploiting the notion of Monge solution and using it to study discontinuous eikonal equations in metric measure spaces which produces even new results in the Euclidean spaces. Second, constructing a two-person continuous-time game in a geodesic space and show that the value function is the unique solution of the Hamilton-Jacobi equation. Our result develops, in a general geometric setting, the classical connection between differential games and the viscosity solutions to possibly nonconvex Hamilton-Jacobi equations. Third, using first-order and second-order PDE-based approaches to study the horizontally quasiconvex (h-quasiconvex for short) functions in the Heisenberg group and apply the characterizations to construct h-quasiconvex envelope and study the h-convexity preserving property for horizontal curvature flow in the Heisenberg group.

Academic Significance and Societal Importance of the Research Achievements

Although several notions of viscosity solutions to the HJ equations on metric spaces have been introduced, our research reveals intrinsic connections between numerous results on HJ equations in general settings and has great potential to be applied in other fields.

Report

(5 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • Research Products

    (23 results)

All 2024 2023 2022 2021 2020 Other

All Int'l Joint Research (3 results) Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 3 results) Presentation (13 results) (of which Int'l Joint Research: 8 results,  Invited: 9 results) Remarks (3 results)

  • [Int'l Joint Research] University of Cincinnati(米国)

    • Related Report
      2023 Annual Research Report
  • [Int'l Joint Research] University of Cincinnati(米国)

    • Related Report
      2022 Research-status Report
  • [Int'l Joint Research] University of Cincinnati(米国)

    • Related Report
      2020 Research-status Report
  • [Journal Article] Horizontally quasiconvex envelope in the Heisenberg group2023

    • Author(s)
      Kijowski Antoni、Liu Qing、Zhou Xiaodan
    • Journal Title

      Revista Matematica Iberoamericana

      Volume: in press Issue: 1 Pages: 57-92

    • DOI

      10.4171/rmi/1417

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Differential games and Hamilton-Jacobi-Isaacs equations in metric spaces2022

    • Author(s)
      Qing Liu and Xiaodan Zhou
    • Journal Title

      Minimax Theory and its Applications

      Volume: in press

    • Related Report
      2021 Research-status Report
  • [Journal Article] Equivalence of solutions of eikonal equation in metric spaces2021

    • Author(s)
      Qing Liu, Nageswari Shanmugalingam, Zhou Xiaodan
    • Journal Title

      Journal of Differential Equations

      Volume: 272 Pages: 979-1014

    • DOI

      10.1016/j.jde.2020.10.018

    • Related Report
      2020 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Horizontal convex envelope in the Heisenberg group and applications to sub-elliptic equations2021

    • Author(s)
      Liu Qing、Zhou Xiaodan
    • Journal Title

      ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE

      Volume: XXII Pages: 30-30

    • DOI

      10.2422/2036-2145.201907_001

    • Related Report
      2020 Research-status Report
    • Peer Reviewed
  • [Presentation] Discontinuous eikonal equations in metric measure spaces2024

    • Author(s)
      Xiaodan Zhou
    • Organizer
      AMS Spring Eastern Sectional Meeting
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Eikonal equations in metric spaces2024

    • Author(s)
      Xiaodan Zhou
    • Organizer
      Smith College Undergraduate seminar
    • Related Report
      2023 Annual Research Report
    • Invited
  • [Presentation] Hamilton-Jacobi equations in metric measure spaces2023

    • Author(s)
      Xiaodan Zhou
    • Organizer
      OIST Analysis and Partial Differential Equations Summer School
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Discontinuous eikonal equations in metric measure spaces2023

    • Author(s)
      Xiaodan Zhou
    • Organizer
      OIST Workshop Geometric Aspects of Partial Differential Equations
    • Related Report
      2023 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Discontinuous eikonal equations in metric measure spaces2022

    • Author(s)
      Xiaodan Zhou
    • Organizer
      MATRIX-RIMS Tandem Workshop on Geometric Analysis in Harmonic Analysis and PDE
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] Eikonal equations on metric measure spaces2022

    • Author(s)
      Xiaodan Zhou
    • Organizer
      RIMS Women in Mathematics
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] Horizontally quasiconvex envelope in the Heisenberg group2022

    • Author(s)
      Xiaodan Zhou
    • Organizer
      The 47th Sapporo Symposium on Partial Differential Equations
    • Related Report
      2022 Research-status Report
    • Int'l Joint Research
  • [Presentation] Characterization of horizontal quasiconvexity in the Heisenberg group and applications2022

    • Author(s)
      Xiaodan Zhou
    • Organizer
      PDE and Analysis Seminar, University of Pittsburgh
    • Related Report
      2022 Research-status Report
    • Invited
  • [Presentation] Horizontal quasiconvex envelope in the Heisenberg group2021

    • Author(s)
      Xiaodan Zhou
    • Organizer
      Himeji conference of Partial Differential Equation
    • Related Report
      2021 Research-status Report
    • Int'l Joint Research
  • [Presentation] Horizontally quasiconvex envelope in the Heisenberg group2021

    • Author(s)
      Xiaodan Zhou
    • Organizer
      Geometric and functional inequalities and applications seminar
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Horizontally quasiconvex envelope in the Heisenberg group2021

    • Author(s)
      Xiaodan Zhou
    • Organizer
      PDE seminar of HKST
    • Related Report
      2021 Research-status Report
    • Invited
  • [Presentation] Eikonal Equations on Metric Spaces2020

    • Author(s)
      Xiaodan Zhou
    • Organizer
      Partial Differential Equations under Various Metrics
    • Related Report
      2020 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Horizontal convex envelope in the Heisenberg group2020

    • Author(s)
      Xiaodan Zhou
    • Organizer
      Chinese Academy of Sciences seminar
    • Related Report
      2020 Research-status Report
    • Invited
  • [Remarks] DISCONTINUOUS EIKONAL EQUATIONS Preprint

    • URL

      https://arxiv.org/pdf/2308.06872

    • Related Report
      2023 Annual Research Report
  • [Remarks] A 2nd-ORDER OPERATOR FOR H-QUASICONVEXITY Preprint

    • URL

      https://arxiv.org/pdf/2312.10364

    • Related Report
      2023 Annual Research Report
  • [Remarks] Hamilton-Jacobi Equation in metric spaces

    • URL

      https://arxiv.org/abs/2308.08073

    • Related Report
      2023 Annual Research Report

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Published: 2020-09-29   Modified: 2025-01-30  

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