Creation and development of geometric analysis to connect discrete geometry and smooth geometry, and collaborations with materials science

• Project Area: Discrete Geometric Analysis for Materials Design
• Project/Area Number: 18H04487
• Research Category: Grant-in-Aid for Scientific Research on Innovative Areas (Research in a proposed research area)
• Allocation Type: Single-year Grants
• Research Institution: Kyushu University
• Principal Investigator: 小磯 深幸 九州大学, マス・フォア・インダストリ研究所, 教授 (10178189)
• Project Period (FY): 2018-04-01 – 2020-03-31
• Project Status: Completed (Fiscal Year 2019)
• Budget Amount: ¥7,800,000 (Direct Cost: ¥6,000,000、Indirect Cost: ¥1,800,000); Fiscal Year 2019: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000); Fiscal Year 2018: ¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
• Keywords: 変分問題 / ウルフ図形 / 非等方的エネルギー / エネルギー極小解 / 自由境界問題 / 特異点 / 離散幾何 / 平均曲率 / 幾何解析 / 非等方的平均曲率流 / Wulff図形

Outline of Annual Research Achievements

曲面の非等方的エネルギーの変分問題は、エネルギー密度関数の適切な選択により、液体・方向性のある液晶・結晶(固体)の全ての最適形状を求める一般的方法を与えると考えられる。解としては、滑らかな曲面・カドのある曲面・多面体（あるいは離散曲面）が現れる。これらを統一的に扱うことにより、滑らかな微分幾何学や楕円型作用素が支配的な曲面の研究では扱えなかった曲面の解析及びエネルギー勾配流の研究を行うことが可能となると期待される。
2018年度には、「区分的Cr級弱はめ込み」という一般次元ユークリッド空間内の超曲面のクラスを定義し、このクラスに属する超曲面について、非等方的エネルギーの臨界点の幾何学的及び解析的な性質、いくつかの有用な積分公式、エネルギー汎関数が凸性を持つ場合に対する閉超曲面についてのエネルギー極小解の一意性定理、エネルギー汎関数に凸性を仮定しない場合の「良い性質」を持つ臨界点の非一意性定理を証明した。ただしここでは非等方的エネルギー密度関数に微分可能性を仮定していた。しかしながら、結晶の数理モデルを扱うためには、ウルフ図形（同じ体積を囲む超閉曲面の中での非等方的エネルギーの最小解）が多面体の場合、即ち非等方的エネルギー密度関数が微分不可能な点を持つ場合を考える必要がある。本年度は、このような場合について、いくつかの付加的条件のもとで、非等方的エネルギー極小な閉超曲面はウルフ図形に限ることを証明した。その際、区分的に滑らかな曲線・曲面に対し、法ベクトルや曲率などの幾何学量を定義し、古典的なシュタイナー公式並びにミンコフスキー公式が成立することを見た。
また、非等方的エネルギー密度関数が2階連続的微分可能で凸性を持つ場合については、(n+1)次元ユークリッド空間の楔状閉領域あるいは錐状閉領域における超曲面についての自由境界問題のエネルギーの極小解の一意性定理を得た。

Research Progress Status

令和元年度が最終年度であるため、記入しない。

Strategy for Future Research Activity

令和元年度が最終年度であるため、記入しない。

Research Products

• [Journal Article] Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and self-similar solutions of anisotropic mean curvature flow (2021)
  • Author(s): Miyuki Koiso
  • Journal Title: Springer Proceedings in Mathematics & Statistics Volume: 349 Pages: 169-185
  • DOI: 10.1007/978-3-030-68541-6_10
  • ISBN: 9783030685409, 9783030685416
  • Peer Reviewed
• [Journal Article] Uniqueness for closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (2020)
  • Author(s): Miyuki Koiso
  • Journal Title: RIMS Kokyuroku Volume: 2146 Pages: 75-85
  • NAID: 120006888288
  • Open Access
• [Journal Article] Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature (2020)
  • Author(s): Miyuki Koiso
  • Journal Title: Advanced Studies in Pure Mathematics Volume: 85 Pages: 239-258
  • DOI: 10.2969/aspm/08510239
  • Peer Reviewed
• [Journal Article] Heinz-type mean curvature estimates in Lorentz-Minkowski space (2020)
  • Author(s): Honda Atsufumi、Kawakami Yu、Koiso Miyuki、Tori Syunsuke
  • Journal Title: Revista Matematica Complutense Volume: - Issue: 3 Pages: 641-651
  • DOI: 10.1007/s13163-020-00373-9
  • Peer Reviewed / Open Access
• [Journal Article] Stability of helicoidal surfaces with constant mean curvature (2020)
  • Author(s): Yuta Hatakeyama、Miyuki Koiso
  • Journal Title: International Journal of Mathematics for Industry Volume: 12 Issue: 01 Pages: 2050007-2050007
  • DOI: 10.1142/s2661335220500070
  • Peer Reviewed / Open Access
• [Journal Article] Uniqueness of Closed Equilibrium Hypersurfaces for Anisotropic Surface Energy and Application to a Capillary Problem (2019)
  • Author(s): Miyuki Koiso
  • Journal Title: Mathematical and Computational Applications Volume: 24 Issue: 4 Pages: 88-88
  • DOI: 10.3390/mca24040088
  • Peer Reviewed / Open Access
• [Journal Article] Geometry of anisotropic surface energy (2018)
  • Author(s): Miyuki Koiso
  • Journal Title: RIMS Kokyuroku Volume: 2068 Pages: 57-68
  • Open Access
• [Journal Article] Fold singularities on spacelike CMC surfaces in Lorentz-Minkowski space (2018)
  • Author(s): HONDA Atsufumi、KOISO Miyuki、SAJI Kentaro
  • Journal Title: Hokkaido Mathematical Journal Volume: 47 Issue: 2 Pages: 245-267
  • DOI: 10.14492/hokmj/1529308818
  • Peer Reviewed
• [Journal Article] On bifurcation and local rigidity of triply periodic minimal surfaces in R^3 (2018)
  • Author(s): Koiso Miyuki、Piccione Paolo、Shoda Toshihiro
  • Journal Title: Annales de l'Institut Fourier Volume: 68 Issue: 6 Pages: 2743-2778
  • DOI: 10.5802/aif.3222
  • Peer Reviewed / Int'l Joint Research
• [Presentation] On the bifurcation and stability for surfaces with constant mean curvature bounded by two coaxial circles (2021)
  • Author(s): 奥田健斗，小磯深幸
  • Organizer: 日本応用数理学会第17回研究部会連合発表会
• [Presentation] Uniqueness of local minimizers for crystalline variational problems (2021)
  • Author(s): Kento Okuda and Miyuki Koiso
  • Organizer: The 27th Osaka City University International Academic Symposium
  • Int'l Joint Research / Invited
• [Presentation] Stable anisotropic capillary hypersurfaces in a wedge (2020)
  • Author(s): Miyuki Koiso
  • Organizer: Mini-symposium : Nonlinear Geometric Partial Differential Equations (Okinawa Institute of Science and Technology)
  • Int'l Joint Research / Invited
• [Presentation] Stable anisotropic capillary hypersurfaces in a wedge (2020)
  • Author(s): Miyuki Koiso
  • Organizer: Workshop and School on Geometric Analysis and Discrete Geometry (KIAS, Korea)
  • Int'l Joint Research / Invited
• [Presentation] Geometry of piecewise-continuous curves and surfaces (2020)
  • Author(s): Miyuki Koiso
  • Organizer: The closing workshop of the project "Geometric Shape Generation", (Tokyo Institute of Technology)
  • Int'l Joint Research
• [Presentation] クリスタライン変分問題に対するエネルギー極小解の一意性について (2020)
  • Author(s): 小磯深幸，奥田健斗
  • Organizer: 日本応用数理学会2020年度年会
• [Presentation] 区分的に滑らかな非等方的平均曲率一定超曲面の不安定性について (2020)
  • Author(s): 軸丸芳揮，小磯深幸
  • Organizer: 日本応用数理学会2020年度年会
• [Presentation] Lorentz-Minkowski空間におけるHeinz型の平均曲率の評価について (2020)
  • Author(s): 川上裕，本田淳史，小磯深幸，通峻祐
  • Organizer: 日本数学会2020年度秋季総合分科会
• [Presentation] Anisotropic surface energy and crystalline variational problems (2019)
  • Author(s): Miyuki Koiso
  • Organizer: 9th International Congress in Industrial and Applied Mathematics
  • Int'l Joint Research
• [Presentation] Variational problem for anisotropic surface energy (2019)
  • Author(s): Miyuki Koiso
  • Organizer: Geometric Analysis and General Relativity (University of Tokyo)
  • Int'l Joint Research / Invited
• [Presentation] Variational problem for anisotropic surface energy (2019)
  • Author(s): Miyuki Koiso and Yoshiki Jikumaru
  • Organizer: Materials Research Meeting 2019 (MRM2019)
  • Int'l Joint Research
• [Presentation] 特異点を許容する超曲面に対する非等方的エネルギーの変分問題 (2019)
  • Author(s): 小磯深幸
  • Organizer: 第20回特異点研究会「特異点と時空、および関連する物理」，(2019年 1月12日（土）～ 1月14日（月・祝），九州大学 西新プラザ)
  • Invited
• [Presentation] Towards crystalline variational problems from elliptic variational problems (2019)
  • Author(s): Miyuki Koiso
  • Organizer: Introductory workshop on discrete differential geometry (January 21-24, 2019, Korea University, Seoul)
  • Int'l Joint Research / Invited
• [Presentation] 特異点を持つ非等方的平均曲率一定閉超曲面の一意性について (2019)
  • Author(s): 小磯深幸
  • Organizer: 淡路島幾何学研究集会2019，(2019年1月25日(金)-1月27日(日)，阿那賀地区公民館，淡路島・南あわじ市)
• [Presentation] Variational problems of anisotropic surface energy for hypersurfaces with singular points (2019)
  • Author(s): Miyuki Koiso
  • Organizer: AMS Spring Central and Western Joint Sectional Meeting, (University of Hawaii at Manoa, Honolulu, HI, March 22-24, 2019),USA
  • Int'l Joint Research
• [Presentation] Uniqueness for closed embedded non-smooth hypersurfaces with constant anisotropic mean curvature (2018)
  • Author(s): 小磯深幸
  • Organizer: 京都大学数理解析研究所（RIMS）共同研究・公開型「偏微分方程式の解の形状解析」
  • Invited
• [Presentation] Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature (2018)
  • Author(s): Miyuki Koiso
  • Organizer: The 10th Mathematical Society of Japan Seasonal Institute (MSJ-SI): The Role of Metrics in the Theory of Partial Differential Equations (from July 2 to 13, 2018, Hokkaido University)
  • Int'l Joint Research / Invited
• [Presentation] 結晶に纏わる（微分）幾何学 (2018)
  • Author(s): Miyuki Koiso
  • Organizer: Mini-Workshop on Geometry and Mathematical Science, Osaka City Univeristy
  • Int'l Joint Research / Invited
• [Presentation] Crystalline variational problem and applications to capillary problems (2018)
  • Author(s): Miyuki Koiso
  • Organizer: 7th International Conference on Mathematical Modeling in Physical Sciences (27-31 August, 2018, Moscow), Russia
  • Int'l Joint Research / Invited
• [Presentation] Uniqueness problem for closed non-smooth hypersurfaces with constant anisotropic mean curvature and applications to anisotropic mean curvature flow (2018)
  • Author(s): Miyuki Koiso
  • Organizer: Conference “Analysis and Geometry in Minimal Surface Theory”(from 3rd Dec. to 7th Dec., 2018, Hilton Gyeongju), Korea
  • Int'l Joint Research / Invited
• [Remarks] 九州大学マス・フォア・インダストリ研究所 所員紹介詳細 小磯深幸/教授
  • URL: https://www.imi.kyushu-u.ac.jp/academic_staffs/view/86
• [Remarks] Institute of Mathematics for Industry, Kyushu U.
  • URL: https://www.imi.kyushu-u.ac.jp/eng/academic_staffs/view/85
• [Remarks] Institute of Mathematics for Industry, KOISO
  • URL: https://www.imi.kyushu-u.ac.jp/eng/academic_staffs/view/85
• [Funded Workshop] Summer School 2018 "Geometric Shape Generation" (2018)