Researcher Database

Nao Hamamuki
Faculty of Science Mathematics Mathematics
Associate Professor

Researcher Profile and Settings

Affiliation

  • Faculty of Science Mathematics Mathematics

Job Title

  • Associate Professor

Degree

  • Doctor of Philosophy in the field of Mathematical Sciences(The University of Tokyo)

Research funding number

  • 70749754

J-Global ID

Research Interests

  • Viscosity solution   Hamilton-Jacobi equation   Comparison principle   Nonlinear partial differential equations   Level set method   

Research Areas

  • Natural sciences / Mathematical analysis / Nonlinear partial differential equations

Academic & Professional Experience

  • 2016/10 - Today Hokkaido University Department of Mathematics Associate Professor
  • 2015/02 - 2016/09 Hokkaido University Department of Mathematics Assistant Professor
  • 2014/04 - 2015/01 Waseda University Faculty of Education and Integrated Arts and Sciences JSPS Research Fellowship PD
  • 2013/10 - 2014/03 The University of Tokyo Graduate School of Mathematical Sciences JSPS Research Fellowship PD

Education

  • 2009/04 - 2013/09  The University of Tokyo  Graduate School of Mathematical Sciences

Association Memberships

  • THE MATHEMATICAL SOCIETY OF JAPAN   

Research Activities

Published Papers

  • An improvement of level set equations via approximation of a distance function
    N. Hamamuki
    Appl. Anal. 98 (10) 1901 - 1915 2019 [Refereed][Not invited]
  • On a dynamic boundary condition for singular degenerate parabolic equations in a half space
    Y. Giga, N. Hamamuki
    NoDEA Nonlinear Differential Equations Appl. 25 (6) 1 - 39 2018 [Refereed][Not invited]
  • Two approaches to an approximation of a distance function to moving interfaces
    N. Hamamuki
    Oberwolfach Rep. 14 (1) 297 - 299 2017 [Not refereed][Not invited]
  • Nao Hamamuki
    JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 438 (1) 184 - 199 0022-247X 2016/06 [Refereed][Not invited]
     
    We present a new Harnack inequality for non-negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices. This estimate applies to all supersolutions and has the Harnack constant depending on the graph distance on lattices: For the proof we modify the proof of the weak Harnack inequality. Applying the same idea to elliptic equations in a Euclidean space, we also derive a Harnack type inequality for non-negative viscosity supersolutions. (C) 2016 Elsevier Inc. All rights reserved.
  • Nao Hamamuki, Eleftherios Ntovoris
    INTERFACES AND FREE BOUNDARIES 18 (4) 579 - 621 1463-9963 2016 [Refereed][Not invited]
     
    In this paper we set up a rigorous justification for the reinitialization algorithm. Using the theory of viscosity solutions, we propose a well-posed Hamilton-Jacobi equation with a parameter, which is derived from homogenization for a Hamiltonian discontinuous in time which appears in the reinitialization. We prove that, as the parameter tends to infinity, the solution of the initial value problem converges to a signed distance function to the evolving interfaces. A locally uniform convergence is shown when the distance function is continuous, whereas a weaker notion of convergence is introduced to establish a convergence result to a possibly discontinuous distance function. In terms of the geometry of the interfaces, we give a necessary and sufficient condition for the continuity of the distance function. We also propose another simpler equation whose solution has a gradient bound away from zero.
  • Nao Hamamuki, Atsushi Nakayasu, Tokinaga Namba
    JOURNAL OF DIFFERENTIAL EQUATIONS 259 (11) 6672 - 6693 0022-0396 2015/12 [Refereed][Not invited]
     
    We study a cell problem arising in homogenization for a Hamilton-Jacobi equation whose Hamiltonian is not coercive. We introduce a generalized notion of effective Hamiltonians by approximating the equation and characterize the solvability of the cell problem in terms of the generalized effective Hamiltonian. Under some sufficient conditions, the result is applied to the associated homogenization problem. We also show that homogenization for non-coercive equations fails in general. (C) 2015 Elsevier Inc. All rights reserved.
  • An improved level set method for Hamilton-Jacobi equations
    N. Hamamuki
    RIMS Kokyuroku 1962 27 - 44 2015 [Not refereed][Not invited]
  • EIKONAL EQUATIONS IN METRIC SPACES
    Yoshikazu Giga, Nao Hamamuki, Atsushi Nakayasu
    TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY 367 (1) 49 - 66 0002-9947 2015/01 [Refereed][Not invited]
     
    A new notion of a viscosity solution for Eikonal equations in a general metric space is introduced. A comparison principle is established. The existence of a unique solution is shown by constructing a value function of the corresponding optimal control theory. The theory applies to infinite dimensional setting as well as topological networks, surfaces with singularities.
  • Nao Hamamuki
    DISCRETE & COMPUTATIONAL GEOMETRY 52 (2) 221 - 239 0179-5376 2014/09 [Refereed][Not invited]
     
    We establish an isoperimetric inequality with constraint by n-dimensional lattices. We prove that, among all sets which consist of lattice translations of a given rectangular parallelepiped, a cube is the best shape to minimize the ratio involving its perimeter and volume as long as the cube is realizable by the lattice. For its proof a solvability of finite difference Poisson-Neumann problems is verified. Our approach to the isoperimetric inequality is based on the technique used in a proof of the Aleksandrov-Bakelman-Pucci maximum principle, which was originally proposed by Cabr (Butll Soc Catalana Mat 15: 7-27, 2000) to prove the classical isoperimetric inequality.
  • ASYMPTOTICALLY SELF-SIMILAR SOLUTIONS TO CURVATURE FLOW EQUATIONS WITH PRESCRIBED CONTACT ANGLE AND THEIR APPLICATIONS TO GROOVE PROFILES DUE TO EVAPORATION-CONDENSATION
    Nao Hamamuki
    ADVANCES IN DIFFERENTIAL EQUATIONS 19 (3-4) 317 - 358 1079-9389 2014/03 [Refereed][Not invited]
     
    We study the asymptotic behavior of solutions to fully nonlinear second order parabolic equations including a generalized curvature flow equation which was introduced by Mullins in 1957 as a model of evaporation-condensation. We prove that, in the multi-dimensional half space, solutions of the problem with prescribed contact angle asymptotically converge to a self-similar solution of the associated problem under a suitable rescaling. Several properties of the profile function of the self-similar solution are also investigated. We show that the profile function has a corner and that the angles are determined by points at which the equation is degenerate. We also study the depth of the groove, which is represented by the value of the profile function at the boundary. Among other results it turns out that, as the contact angle tends to zero, the depth of the groove is well approximated by the linearized problem.
  • On large time behavior of Hamilton-Jacobi equations with discontinuous source terms
    N. Hamamuki
    GAKUTO Internat. Ser. Math. Sci. Appl. 36 83 - 112 2013 [Refereed][Not invited]
  • Yoshikazu Giga, Nao Hamamuki
    Communications in Partial Differential Equations 38 (2) 199 - 243 0360-5302 2013 [Refereed][Not invited]
     
    We study the initial-value problem for a Hamilton-Jacobi equation whose Hamiltonian is discontinuous with respect to state variables. Our motivation comes from a model describing the two dimensional nucleation in crystal growth phenomena. A typical equation has a semicontinuous source term. We introduce a new notion of viscosity solutions and prove among other results that the initial-value problem admits a unique global-in-time uniformly continuous solution for any bounded uniformly continuous initial data. We also give a representation formula of the solution as a value function by the optimal control theory with a semicontinuous running cost function. © 2013 Copyright Taylor and Francis Group, LLC.
  • A class of nowhere differentiable functions satisfying some concavity-type estimate
    Y. Fujita, N. Hamamuki, A. Siconolfi, N. Yamaguchi
    Acta Math. Hungar. (accepted) [Refereed][Not invited]

Conference Activities & Talks

  • On large time behavior of some crystal growth problems  [Invited]
    HAMAMUKI Nao
    表面・界面ダイナミクスの数理18  2019/10
  • Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms  [Invited]
    HAMAMUKI Nao
    9th International Congress on Industrial and Applied Mathematics - ICIAM 2019  2019/07
  • A comparison principle for viscosity solutions of a boundary value problem without the normal derivative  [Invited]
    HAMAMUKI Nao
    京都大学NLPDEセミナー  2019/06
  • Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms  [Invited]
    HAMAMUKI Nao
    名古屋微分方程式セミナー  2019/04
  • Asymptotic shape of solutions to the mean curvature flow equation with discontinuous source terms  [Invited]
    HAMAMUKI Nao
    微分方程式と逆問題をめぐって  2019/03
  • 不連続外力項を持つ曲率流方程式の粘性解について  [Invited]
    HAMAMUKI Nao
    第一回はこだて数理解析研究集会  2018/11
  • On a dynamic boundary value problem of the level-set mean curvature flow equation  [Invited]
    HAMAMUKI Nao
    Advanced Developments for Surface and Interface Dynamics - Analysis and Computation  2018/06
  • On a dynamic boundary condition for singular degenerate parabolic equations in a half space  [Not invited]
    HAMAMUKI Nao
    日本数学会2018年度年会  2018/03
  • A discrete game interpretation for a dynamic boundary value problem of the mean curvature flow equation  [Invited]
    HAMAMUKI Nao
    京都大学NLPDEセミナー  2018/01
  • On viscosity solutions in metric spaces  [Invited]
    HAMAMUKI Nao
    離散幾何構造セミナー  2017/12
  • On a dynamic boundary condition for singular degenerate parabolic equations  [Invited]
    HAMAMUKI Nao
    解析セミナー  2017/12
  • A discrete game interpretation for a dynamic boundary value problem of the mean curvature flow equation  [Invited]
    HAMAMUKI Nao
    偏微分方程式セミナー  2017/12
  • 界面発展方程式の動的境界値問題について  [Invited]
    HAMAMUKI Nao
    応用数学に関する勉強会(応用数学セミナー)@芝浦工大  2017/11
  • 均質化問題と粘性解理論  [Invited]
    HAMAMUKI Nao
    日本応用数理学会2017年度年会  2017/09
  • On surface evolutions under some dynamic boundary conditions  [Invited]
    HAMAMUKI Nao
    Nonlinear PDE for Future Applications -Optimal Control and PDE-  2017/07
  • A comparison principle for singular degenerate parabolic equations under some dynamic boundary conditions  [Invited]
    HAMAMUKI Nao
    5th Italian-Japanese Workshop on Geometric Properties for Parabolic and Elliptic PDE's  2017/05
  • 粘性解に対する均質化問題ー不連続方程式への拡張とその応用ー  [Invited]
    HAMAMUKI Nao
    非線形現象の数値シミュレーションと解析2017  2017/03
  • 不連続な加法的固有値問題に対する粘性解とその応用  [Invited]
    HAMAMUKI Nao
    金沢解析セミナー  2017/03
  • Two approaches to an approximation of a distance function to moving interfaces  [Invited]
    HAMAMUKI Nao
    第18回北東数学解析研究会  2017/02
  • Two approaches to an approximation of a distance function to moving interfaces  [Invited]
    HAMAMUKI Nao
    Emerging Developments in Interfaces and Free Boundaries  2017/01
  • Harnack inequalities for supersolutions of fully nonlinear elliptic difference equations  [Invited]
    HAMAMUKI Nao
    Towards regularity  2016/09
  • On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and its application to homogenization problems  [Invited]
    HAMAMUKI Nao
    The 11th AIMS Conference on Dynamical Systems, Differential Equations and Applications  2016/07
  • A discrete isoperimetric inequality on lattices  [Invited]
    HAMAMUKI Nao
    Hamilton-Jacobi Equations:New trends and applications  2016/05

Awards & Honors

  • 2016/02 Inoue Foundation for Science 32nd Inoue Research Award for Young Scientists
     
    受賞者: HAMAMUKI Nao
  • 2014/02 Japan Society for the Promotion of Science 4th Ikushi Prize
     Crystal Growth Phenomena and Hamilton-Jacobi Equations 
    受賞者: HAMAMUKI Nao
  • 2013/09 Mathematical Society of Japan Takebe Katahiro Prize for Encouragement of Young Researchers
     Mathematical analysis for Hamilton-Jacobi equations and its application to crystal growth phenomena 
    受賞者: HAMAMUKI Nao

Research Grants & Projects

  • 特異構造を持つ界面発展方程式と境界値問題
    Japan Society for the Promotion of Science:Grant-in-Aid for Early-Career Scientists
    Date (from‐to) : 2019/04 -2023/03 
    Author : HAMAMUKI Nao
  • Analysis of boundary value problems for fully nonlinear partial differential equations and its applications
    Inamori Foundation:Research Grants
    Date (from‐to) : 2018/04 -2019/03 
    Author : HAMAMUKI Nao
  • 離散と連続をつなぐ粘性解理論の構築
    Japan Society for the Promotion of Science:Grant-in-Aid for Young Scientists (B)
    Date (from‐to) : 2016/04 -2019/03 
    Author : HAMAMUKI Nao
  • 界面ダイナミクスの数学解析に向けた粘性解理論の深化
    The Sumitomo Foundation:Grant for Basic Science Research Projects
    Date (from‐to) : 2015/11 -2016/10 
    Author : HAMAMUKI Nao
  • 粘性解理論とその材料科学分野への応用
    Japan Society for the Promotion of Science:Grant-in-Aid for JSPS Research Fellow PD
    Date (from‐to) : 2014/04 -2015/01 
    Author : HAMAMUKI Nao
  • 結晶成長現象とハミルトン・ヤコビ方程式
    Japan Society for the Promotion of Science:Grant-in-Aid for JSPS Research Fellow DC1
    Date (from‐to) : 2011/04 -2014/03 
    Author : HAMAMUKI Nao

Educational Activities

Teaching Experience

  • Analytic Studies
    開講年度 : 2018
    課程区分 : 修士課程
    開講学部 : 理学院
    キーワード : 関数の微分可能性、非線形偏微分方程式、弱微分、最大値原理
  • Analysis B
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 常微分方程式、解の存在、解の一意性
  • Advanced Mathematical Analysis
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 関数の微分可能性、非線形偏微分方程式、弱微分、最大値原理
  • Calculus I
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 全学教育
    キーワード : 数列,収束,関数,極限,微分,偏微分,テイラーの定理
  • Exercises on Basic Mathematics C
    開講年度 : 2018
    課程区分 : 学士課程
    開講学部 : 理学部
    キーワード : 実数、数列、収束、連続、微分、積分


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