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2012 Fiscal Year Final Research Report

Diffusion and Geometry of Domain

Research Project

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Project/Area Number 20340031
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionTohoku University (2012)
Hiroshima University (2008-2011)

Principal Investigator

SAKAGUCHI Shigeru  東北大学, 大学院・情報科学研究科, 教授 (50215620)

Co-Investigator(Kenkyū-buntansha) MISAWA Masashi  熊本大学, 大学院・自然科学研究科, 教授 (40242672)
MIKAMI Toshio  広島大学, 大学院・工学研究院, 教授 (70229657)
SHIBATA Tetsutaro  広島大学, 大学院・工学研究院, 教授 (90216010)
KANON Yukio  愛媛大学, 教育学部, 教授 (00177776)
Co-Investigator(Renkei-kenkyūsha) KAWAKAMI Tatsuki  大阪府立大学, 大学院・工学研究科, 講師 (20546147)
ISHIGE Kazuhiro  東北大学, 大学院・理学研究科, 教授 (90272020)
OGAWA Takayoshi  東北大学, 大学院・理学研究科, 教授 (20224107)
Project Period (FY) 2008 – 2012
Keywords熱方程式 / 非線形拡散方程式 / 非線形楕円型方程式 / 不変な等位面 / 領域の幾何 / 球面の特徴付け / 超平面の特徴付け / 円柱面の特徴付け
Research Abstract

Partial differential equations describing diffusion phenomena have been widely considered. To know the relationship between the behavior of solutions and the geometry of domain, we showed both the relationship between the initial behavior and the curvatures of the boundary and that between the existence of a stationary level surface with time and the symmetry of domain. In particular, we obtained characterizations of the sphere, the hyperplane, and the circular cylinder involving a stationary level surface. These yielded a new development of inverse problems determining the geometry of domain. Also, as a by-product, we obtained Liouville-type theorems for viscosity solutions of fully nonlinear elliptic partial differential equations describing an important class of Weingarten hypersurfaces.

  • Research Products

    (34 results)

All 2013 2012 2011 2010 2009 2008 Other

All Journal Article (11 results) (of which Peer Reviewed: 10 results) Presentation (20 results) Book (2 results) Remarks (1 results)

  • [Journal Article] Thep-Laplacian Operator and Applications2013

    • Author(s)
      Shigeru Sakaguchi, Fast diffusion and geometry of domain, in
    • Journal Title

      Oberwolfach Report

      Volume: (印刷中)

    • URL

      http://www.ems-ph.org/journals/journal.php?jrn=OWR

  • [Journal Article] Matzoh ball soup revisited: t he boundary regularity issue2013

    • Author(s)
      Rolando Magnanini and Shigeru Sak aguchi
    • Journal Title

      Mathematical Methods in Applied Sciences

      Volume: (掲載受理印刷中)

    • DOI

      DOI:10.1002/mma.1551

    • Peer Reviewed
  • [Journal Article] Stationary level surfaces and Liouville-type theorems characterizing hyperplanes, in "Geometric Properties of Parabolic and Elli ptic PDE's"2013

    • Author(s)
      Shigeru Sakaguchi
    • Journal Title

      Springer INdAM Series

      Volume: Vol.2 Pages: 269-282

    • DOI

      DOI:10.1007/978-88-470-2841-8_17

    • Peer Reviewed
  • [Journal Article] A Holder estimate for nonlinear parabolic systems of p-Laplacian type2013

    • Author(s)
      Masashi Misawa
    • Journal Title

      J. Differential Equati ons

      Volume: 254 Pages: 847-878

    • DOI

      DOI:10.1016/j.jde.2012.10.001

    • Peer Reviewed
  • [Journal Article] A characterization of the Knothe-Rosenblatt processes by a convergence result2012

    • Author(s)
      Toshio Mikami
    • Journal Title

      SIAM J. ControlOptim.

      Volume: 50 Pages: 1903-1920

    • DOI

      DOI:10.1137/100791129

    • Peer Reviewed
  • [Journal Article] Asymptotic behave or of bifurcation curve for sine-Gordon type differential equation2012

    • Author(s)
      Tetsutaro Shibata
    • Journal Title

      Abstract and Applied Analysis

      Volume: 2012 Pages: 16

    • DOI

      DOI:10.1155/2012/753857

    • Peer Reviewed
  • [Journal Article] Interaction between nonlineardiffusion and geometry of domain2012

    • Author(s)
      Rolando Magnanini and Shigeru Sak aguchi
    • Journal Title

      J. Differential Equations

      Volume: 252 Pages: 236-257

    • DOI

      DOI:10.1016/j.jde.2011.08.017

    • Peer Reviewed
  • [Journal Article] Optimization problems on generalclasses of rearrangements2011

    • Author(s)
      F. Cuccu, G. Porru, and S. Sakaguch i
    • Journal Title

      Nonlinear Analysis Theory, Methods & Applicat ions

      Volume: 74 Pages: 5554-5565

    • DOI

      DOI:10.1016/j.na.2011.05.039

    • Peer Reviewed
  • [Journal Article] A Liouville-type theorem for some Weingarten hypersurfaces2011

    • Author(s)
      Shigeru Sakaguchi
    • Journal Title

      Discrete and Continuous Dynamical Systems -Series S

      Volume: 4 Pages: 887-895

    • DOI

      DOI:10.3934/dcdss.2011.4.887

    • Peer Reviewed
  • [Journal Article] Nonlinear diffusion with abounded stationary level surface2010

    • Author(s)
      Rolando Magnanini and Shigeru Sak aguchi
    • Journal Title

      Annales de l'Inst. Henri Poincare (C) Analyse Non Lineaire

      Volume: 27 Pages: 937-952

    • DOI

      DOI:10.1016/j.anihpc.2009.12.001

    • Peer Reviewed
  • [Journal Article] Stationary isothermicsurfaces and some characterizationsof the hyperplane in the N-dimension al Euclidean space2010

    • Author(s)
      Rolando Magnanini and Shigeru Sak aguchi
    • Journal Title

      J. Differential Equations

      Volume: 248 Pages: 1112-1119

    • DOI

      DOI:10.1016/j.jde.2009.11.017

    • Peer Reviewed
  • [Presentation] 不変等温面と領域の幾何2013

    • Author(s)
      坂口茂
    • Organizer
      日本数学会年会(解析学賞受賞特別講演)
    • Place of Presentation
      京都大学
    • Year and Date
      2013-03-21
  • [Presentation] Fast diffusion and geometry of domain2013

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      Oberwolfach Mini-Workshop「The p-Laplacian Operator and Applications」
    • Place of Presentation
      ドイツ・Oberwolfach数学研究所
    • Year and Date
      2013-02-14
  • [Presentation] Stationary isothermic surfaces in Euclidean 3-space2012

    • Author(s)
      坂口茂
    • Organizer
      愛媛大学解析セミナー第100回記念講演
    • Place of Presentation
      愛媛大学
    • Year and Date
      2012-11-30
  • [Presentation] Stationary isothermic surfaces and geometry of domain2012

    • Author(s)
      坂口茂
    • Organizer
      第4回名古屋微分方程式研究集会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2012-03-05
  • [Presentation] Nonlinear diffusion and Liouville-type theorems characterizing hyperplanes I, II2012

    • Author(s)
      坂口茂
    • Organizer
      RIMS共同研究「非線形 拡散の数理」
    • Place of Presentation
      京都大学
    • Year and Date
      2012-02-14
  • [Presentation] Stationary isothermic surfaces and Liouville-type theorems characterizing hyperplanes2012

    • Author(s)
      坂口茂
    • Organizer
      RIMS研究集会「偏微分方程式の逆問題解析とその周辺に関する研究」
    • Place of Presentation
      京都大学
    • Year and Date
      2012-01-24
  • [Presentation] 超平面を特徴づける Liouville型定理2011

    • Author(s)
      坂口茂
    • Organizer
      第9回浜松偏微分方程式研究集会
    • Place of Presentation
      静岡大学
    • Year and Date
      2011-12-26
  • [Presentation] Stationary isothermic surfaces and Liouville-type theorems characterizing hyperplanes2011

    • Author(s)
      坂口茂
    • Organizer
      Workshop on Differential Geometry and Geometric Analysis
    • Place of Presentation
      東北大学
    • Year and Date
      2011-12-20
  • [Presentation] Stationary level surfaces under low regularity assumption on the domain2011

    • Author(s)
      坂口茂
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      信州大学
    • Year and Date
      2011-09-29
  • [Presentation] Stationary level surfaces under low regularity assumption on the domain2011

    • Author(s)
      坂口茂
    • Organizer
      第7回非線型の諸問題
    • Place of Presentation
      熊本大学
    • Year and Date
      2011-09-24
  • [Presentation] Stationary level surfaces and Liouville-type theorems, Diffusion phenomena and the related topics for maximum points and singularities, The4th MSJ-SI2011

    • Author(s)
      坂口茂
    • Organizer
      Nonlinear Dynamics inPartial Differential Equations
    • Place of Presentation
      九州大学
    • Year and Date
      2011-09-12
  • [Presentation] Diffusion with similar level sets2011

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      BIRS Workshop "Geometric properties of solutions of nonlinear PDEs and their applications"
    • Place of Presentation
      カナダ・バンフ
    • Year and Date
      2011-07-18
  • [Presentation] Level surfaces, Liouville-type theorems and thehyperplane2011

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      2nd Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
    • Place of Presentation
      イタリア・コルトナ
    • Year and Date
      2011-06-22
  • [Presentation] 非線形拡散と領域の幾何2010

    • Author(s)
      坂口茂
    • Organizer
      日本数学会秋季総合分科会
    • Place of Presentation
      名古屋大学
    • Year and Date
      2010-09-23
  • [Presentation] Nonlinear diffusion and geometry of domain, The 8th AIMS Conference on Dynamical Systems2010

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      Differential Equations and Applications
    • Place of Presentation
      ドイツ・ドレスデン工科大学
    • Year and Date
      2010-05-27
  • [Presentation] Nonlinear diffusion and geometry of domain2009

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      RIMS 研究集会「現象解析と関数方程式の新展望」
    • Place of Presentation
      京都大学
    • Year and Date
      2009-11-19
  • [Presentation] A Liouville-type theorem for some Weingarten hypersurfaces2009

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      Nonlinear Evolution Equations and Related Topics
    • Place of Presentation
      早稲田大学
    • Year and Date
      2009-10-10
  • [Presentation] A Liouville-type theorem for some Weingarten hypersurfaces2009

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      1st Italian-Japanese workshop on geometric properties for parabolic and elliptic PDE's
    • Place of Presentation
      東北大学
    • Year and Date
      2009-06-17
  • [Presentation] Diffusion and stationary level surfaces2008

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      第5回世界非線形解析学者会議(WCNA 2008)
    • Place of Presentation
      米国・オーランド
    • Year and Date
      2008-07-05
  • [Presentation] Stationary isothermic surfaces and some characterizations of the hyperplane2008

    • Author(s)
      Shigeru Sakaguchi
    • Organizer
      RIMS研究集会「微分方程式の粘性解とその周辺」
    • Place of Presentation
      京都大学
    • Year and Date
      2008-06-25
  • [Book] Editors, Geometric Properties for Parabolic and Elliptic PDE's, Springer INdAM Series 22013

    • Author(s)
      Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino
    • Total Pages
      1-292
    • Publisher
      Springer
  • [Book] Editors, Geometric Properties for Parabolic and Elliptic PDE's2013

    • Author(s)
      Rolando Magnanini, Shigeru Sakaguchi, Angelo Alvino
    • Total Pages
      1-292
    • Publisher
      Springer INdAM Series 2,Springer
  • [Remarks]

    • URL

      http://researchmap.jp/sigersak2012415/

URL: 

Published: 2014-08-29  

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