Researcher Profile and Settings

Master

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

Affiliation (Master)

• Faculty of Science　Mathematics　Mathematics

researchmap

Profile and Settings

Degree

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

Profile and Settings

• Name (Japanese)

  Hamamuki

• Name (Kana)

  Nao

• Name

  201501088531607716

Achievement

Research Interests

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

Research Areas

• Natural sciences / Mathematical analysis / Nonlinear partial differential equations

Research 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
• 2005/04 - 2009/03  The University of Tokyo  Faculty of Science  Department of Mathematics

Awards

• 2020/02 Hokkaido University President's Award
   

  受賞者: HAMAMUKI Nao
• 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

Published Papers

• A dynamical approach to lower gradient estimates for viscosity solutions of Hamilton-Jacobi equations

  N. Hamamuki, K. Hirose

  SIAM Journal on Mathematical Analysis 55 (4) 3169 - 3204 0036-1410 2023/07 [Refereed][Not invited]
  
• Weak comparison principles for fully nonlinear degenerate parabolic equations with discontinuous source terms

  N. Hamamuki, K. Misu

  Minimax Theory and its Applications 8 (1) 37 - 60 2023 [Refereed][Not invited]
  
• A lower spatially Lipschitz bound for solutions to fully nonlinear parabolic equations and its optimality

  N. Hamamuki, S. Kikkawa

  Indiana University Mathematics Journal 72 (2) 455 - 487 2023 [Refereed][Not invited]
  
• A self-affine property of evolutional type appearing in a Hamilton-Jacobi flow starting from the Takagi function

  Y. Fujita, N. Hamamuki, N. Yamaguchi

  Michigan Mathematical Journal 71 (1) 105 - 120 0026-2285 2022/03 [Refereed][Not invited]
  
• A weak comparison principle and asymptotic behavior of viscosity solutions to the mean curvature flow equation with discontinuous source terms

  N. Hamamuki, K. Misu

  RIMS Kokyuroku 2212 41 - 53 2022 [Not refereed][Invited]
  
• A game-theoretic approach to dynamic boundary problems for level-set curvature flow equations and applications

  N. Hamamuki, Q. Liu

  Partial Differential Equations and Applications 2 (2) 1 - 27 2662-2963 2021/04 [Refereed][Not invited]
   

  This paper is devoted to a game-theoretic approach to the level-set curvature flow equation with nonlinear dynamic boundary conditions. Under the comparison principle for the dynamic boundary problem, we construct a family of deterministic discrete games, whose value functions approximate the unique viscosity solution. We also apply the game approximation to study the convexity preserving properties and the fattening phenomenon for this geometric dynamic boundary problem.
• All the generalized characteristics for the solution to a Hamilton-Jacobi equation with the initial data of the Takagi function

  Y. Fujita, N. Hamamuki, N. Yamaguchi

  Partial Differential Equations and Applications 1 (6) 1 - 20 2662-2963 2020/12 [Refereed][Not invited]
  
• A class of nowhere differentiable functions satisfying some concavity-type estimate

  Y. Fujita, N. Hamamuki, A. Siconolfi, N. Yamaguchi

  Acta Mathematica Hungarica 160 (2) 343 - 359 0236-5294 2020/04 [Refereed][Not invited]
  
• A deterministic game interpretation for fully nonlinear parabolic equations with dynamic boundary conditions

  N. Hamamuki, Q. Liu

  ESAIM: Control, Optimisation and Calculus of Variations 26 (13) 1 - 42 1292-8119 2020 [Refereed][Not invited]
   

  This paper is devoted to deterministic discrete game-theoretic interpretations for fully nonlinear parabolic and elliptic equations with nonlinear dynamic boundary conditions. It is known that the classical Neumann boundary condition for general parabolic or elliptic equations can be generated by including reflections on the boundary to the interior optimal control or game interpretations. We study a dynamic version of such type of boundary problems, generalizing the discrete game-theoretic approach proposed by Kohn-Serfaty (2006, 2010) for Cauchy problems and later developed by Giga-Liu (2009) and Daniel (2013) for Neumann type boundary problems.
• An improvement of level set equations via approximation of a distance function

  N. Hamamuki

  Applicable Analysis 98 (10) 1901 - 1915 0003-6811 2019/07/27 [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 1021-9722 2018 [Refereed][Not invited]
  
• Proceedings of the 42nd Sapporo Symposium on Partial Differential Equations : In memory of Professor Taira Shirota

  Ei S.-I, Giga Y, Hamamuki N, Jimbo S, Kubo H, Ozawa T, Sakajo T, Tonegawa Y, Tsutaya K

  Hokkaido University technical report series in mathematics Department of Mathematics, Hokkaido University 171 1 - 94 1348-4338 2017/08 

  v, 94p
• 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]
  
• Harnack inequalities for supersolutions of fully nonlinear elliptic difference and differential equations

  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.
• A rigorous setting for the reinitialization of first order level set equations

  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.
• On cell problems for Hamilton-Jacobi equations with non-coercive Hamiltonians and their application to homogenization problems

  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 京都大学 1962 27 - 44 1880-2818 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.
• A Discrete Isoperimetric Inequality on Lattices

  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.
• Asymptotically self-similar solutions to curvature flow equations with prescribed contact angle

  N. Hamamuki

  Oberwolfach Rep. 10 (1) 897 - 898 2013 [Not refereed][Not invited]
  
• 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]
  
• Hamilton-Jacobi Equations with Discontinuous Source Terms

  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.

Presentations

• Waiting time effects for the wearing process of a non-convex stone  [Invited]

  HAMAMUKI Nao

  10th International Congress on Industrial and Applied Mathematics - ICIAM 2023  2023/08
• 非凸な石の摩耗過程における待ち時間効果  [Invited]

  HAMAMUKI Nao

  日本応用数理学会2022年度年会  2022/09
• 平均曲率流方程式のゲーム解釈と動的境界値問題  [Invited]

  HAMAMUKI Nao

  日本数学会北海道支部講演会・支部総会  2021/12
• A lower spatially Lipschitz bound for solutions to fully nonlinear parabolic equations and its optimality  [Invited]

  HAMAMUKI Nao

  偏微分方程式セミナー（北海道大学）  2021/10
• A lower spatially Lipschitz bound for solutions to fully nonlinear parabolic equations and its optimality  [Invited]

  HAMAMUKI Nao

  広島微分方程式研究会  2021/10
• Asymptotic behavior of viscosity solutions to the mean curvature flow equation with discontinuous source terms  [Invited]

  HAMAMUKI Nao

  RIMS共同研究（公開型）『偏微分方程式の解の幾何的様相』  2021/06
• Asymptotic behavior of solutions to level-set mean curvature flow equations with discontinuous source terms  [Invited]

  HAMAMUKI Nao

  Interfacial Phenomena in Reaction-Diffusion Systems  2020/08
• Two-dimensional nucleation in crystal growth and large time behavior of solutions  [Invited]

  HAMAMUKI Nao

  The 37th Kyushu Symposium on Partial Differential Equations  2020/01
• On large time behavior of some crystal growth problems  [Invited]

  HAMAMUKI Nao

  Mathematical Aspects of Surface and Interface Dynamics 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（北海道大学）  2017/12
• On a dynamic boundary condition for singular degenerate parabolic equations  [Invited]

  HAMAMUKI Nao

  解析セミナー（神戸大学）  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（北海道大学）  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

Association Memberships

• THE MATHEMATICAL SOCIETY OF JAPAN   

Research Projects

• 界面発展方程式と粘性解の形状解析

  Japan Society for the Promotion of Science：Grant-in-Aid for Scientific Research (C)

  Date (from‐to) : 2023/04 -2027/03 

  Author : HAMAMUKI Nao
• 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