English Research Topics
Partial differential equations, variational problems and eigenvalue problems The theory of partial differential equations is a central area of mathematics, which pervades the natural sciences. My current research involves studying properties of fundamental solutions to elliptic and parabolic partial differential equations, the structure of solutions to nonlinear elliptic equations, and various optimization problems, including eigenvalue optimization problems. I am also interested in various nonlinear phenomena arising in physics, pattern formations in mathematical biology, nonlinear optimization problems and inverse problems.

Recent Publications
  1. (with R. Yamashita) Reconstruction of the Defect by the Enclosure Method for Inverse Problems of the Magnetic Schr\'{o}dinger Operators,
      Tokyo J. Math., 45(2022), 547-577.
  2. (with Y. Osada) Variational problems associated with a system of nonlinear\linebreak Schr\"{o}dinger equations with three wave interaction,
      Discrete Contin. Dyn. Syst. Ser. B 27 (2022), no. 3, 1511-1547.
  3. (with Y. Osada) Asymptotic expansion of the ground state energy for nonlinear Schr\'{o}dinger system with three wave interaction,
      Commun. Pure Appl. Anal. 20 (2021), no. 12, 4239-4251.
  4. (with Y. Ishii) Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs,
      Commun. Pure Appl. Anal. 20 (2021), no. 4, 1633-1679.
  5. (with M.Shibata) Least energy solutions to semi-linear elliptic problems on metric graphs,
      J. Math. Anal. Appl., vol.491 (2020), no.~1, 124297, 22~pp.
  6. (with T, Kajiwara) A Singular Perturbation Problem for heteroclinic Solutions to the FitzHugh-Nagumo Type Reaction-Diffusion System with heterogeneity,
      J. Math. Sci. Univ. Tokyo vol.26 (2019), 141--199.
  7. (with Y. Ishii) Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity,
      Discrete and Conti. Dynamical System, Series-A. vol. 35, no.5(2019), 2807-2875.
  8. (with T. Kajiwara) On a Variational Problem Arising from the Three-component FitzHugh-Nagumo Type Reaction-Diffusion Systems,
      Tokyo J. Math. , 41, no.1 (2018), 131-174. DOI: 10.3836/tjm/1502179257
  9. (with S.Jimbo) Asymptotic behavior of eigenvalues of the Laplacian on a thin domain under the mixed boundary condition,
      Indiana University Mathematics Journal 65(3) 867-898(2016).
  10. (with K. Morimoto) Existence of spike patterns in a chemotaxis model with saturation,
      Discrete Conti. Dynam. System, 31(2011), 139--164.
  11. (with H. Matsuzawa) Multiple stable patterns in a balanced bistable equation with heterogeneous environments,
      Applicable Analysis, to appear
  12. (with Shuichi Kawashima) Hardy type inequality and application to the stability of degenerate stationary waves,
      J. Funct. Anal. Vol.1 (2009), 1--19.
  13. (with Kotaro Morimoto) Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation,
      Comm. Pure Appl. Anal. Vol.7 (2008), 1443-1482.
  14. (with Junping Shi) Optimal Spatial Harvesting Strategy and Symmetry Breaking
      Applied Math. Optim., Vol.58(2008), 89-110.
  15. (with Naoki Shioji) Compactness embedding from W^{1,2}_0(\Omega) to L^{p(x)}(\Omega) and its application to nonlinear elliptic boundary value problem with variable critical exponent,
      J. Math. Anal. Appl., Vol.339(2008),no.2, 1386-1394.
  16. (with Tatsuya Watanabe) A remark on asymptotic profiles of radial solutions with a　vortex to a nonlinear Schroedinger equation
      Comm. Pure Appl. Anal.,Vol.5, No.3, 597--610(2006).
  17. (with T. Ide, K. Tanaka) Multiple stable patterns for some reaction-diffusion equation in disrupted environments,
      Discrete and Continuous Dynamical Systems, vol.14, No.1, 93--116(2006)
  18. (with M. Shibata, T. Watanabe) A symmetry breaking phenomenon and asymptotic profiles of least energy solutions to a nonlinear Schroedinger equation
      Proc. Royal Soc. Edinburgh., Vol.135 A, 357--392(2005)
  19. (with M. Maniwa) A remark on uniqueness of ground state solutions for -△u=K(|x|)|u|^{p-1}u-a(|x|)u in R^n
      Adv. in Math. Sci. Appl., 14 (2004)577--591
  20. (with M. Shibata, S. Sakamoto)On an optimization problem for some nonlinear elliptic boundary value problem
      Applied Math. Optim.50(2004) 259--278.
  21. (with M. Shibata, K. Tada) Existence of positive solutions for some nonlinear elliptic equations on unbounded domains with cylindrical ends
      Nonlinear Analysis 55 (2003)83-101
  22. (with E.M.Harrell II, P.Kroger) On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue
      SIAM J.Math.Anal. 33 (2001), 240--259.
  23. Existence of non-topological solutions for a nonlinear elliptic equation arising from Chern-Simons-Higgs theory in a general background metric
      Diff. Integ. Equ. 14 (2001), 925--935.
  24. (with M. Shibata) On a one-dimensional variational problem related to Cahn-Hilliard energy on a strip-like domain
      Nonlinear Analysis 47 (2001), 1059--1068.
  25. (with S.Chanillo, D.Grieser, M.Imai, I.Ohnishi) Symmetry breaking and other phenomena in an eigenvalue optimization problem
      Comm. Math. Phys.214 (2000), 315--337.
  26. (with S.Chanillo, D.Grieser) The free boundary problm in the optimization of composite membranes
      Contemp. Math. 268 (2000), 61--81.
  27. Existence and semi-classical limit of the least energy solution to a nonlinear Schrodinger equation with electromagnetic fields
      Nonlinear Anal. 41 (2000), 763--778.
  28. (with S. Nishigaki, S. Sugano) Boundedness of integral operators on generalied Morrey spaces and its application to Schroedinger operators
      Proc. Amer. Math. Soc.128 No.4(2000), 1125--1134.
  29. On doubling property for non-negative weak solutions of elliptic and parabolic PDE
      Israel J. Math.115 (2000), 285--302.
  30. A remark on finiteness of the lower spectrum of uniformly elliptic operators with singular potentials
      Integral Eq. and Operator Theory36 (2000), 212--219.
  31. Standing wave solutions for the planer Chern-Simons gauged nonlinear Schrodinger equation with an external elctromagnetic field
      J. Korean Math. Soc. 37 (2000), 977--989.
  32. An estimate on the Heat kernel of magnetic Schrodinger operators and uniformly elliptic operartors with non-negative potentials
      J. London Math. Soc. 62 (2000), 885--903.
  33. (with S. Sugano)Estimates of the fundamental solution for magnetic Schrodinger operators and their applications
      Tohoku Math. J. 52 (2000), 367--382.
  34. (with S. Sugano) Fundamental solution, eigenvalue asymptotics and eigenfunctions of degenerate elliptc operators with positive potentials
      Studia Math. 138(2) (2000), 101--119.
  35. (with S. Sugano) A remark on estimates for uniformly elliptic operators on weighted L^p spaces and Morrey spaces
      Math. Nachr. 209 (2000), 137--150.



Ph. D Thesis
    Regularity and unique continuation property of weak solutions for elliptic partial differential equations of second order
      Ph. D, Tokyo University, 1993.

List of Recent Talks
    Talks at Conference
    1. Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation
        SIAM Conference at SanDiego (USA), December 11, 2011
    2. On a structure of stationary solutions to a nonlinear nonlocal elliptci PDE arising in the Ohmic heating problem
        One Forum Two Cities: Aspects of nonlinear PDEs, National Taiwan University, September 1, 2011
    3. On a structure of stationary solutions to a nonlinear nonlocal elliptci PDE arising in the Ohmic heating problem
        2011 International Workshop on Nonlinear PDE and Application, National Pusan University, June 24, 2011
    4. Multiple stable patterns in a balanced bistable equation with heterogeneous environments,
        7th AIMS International Conference on Dynamical Systems and Differential Equations and Applications, University of Texas at Arlington, May18-21, 2008
    5. Construction and asymptotic behavior of the multi-peak solutions to the Gierer-Meinhardt system with saturation,
        International Conference「First Chile-Japan Workshop on Nonlinear Elliptic and Parabolic PDE」, Santiago, Chile, October 23-26, 2007
    6. Patterns of stationary solutions for a certain reaction-diffusion model in disrupted environments,
        Conference on Nonlinear Elliptic and Parabolic Partial Differential Equations, Jeju, Korea, August 18-20, 2005
    7. Stationary patterns for fourth order equations with an unbalanced bistable nonlinearity under heterogeneous environments
        International Conference「Asymptotic Analysis and Singularities」,at Sendai, July 18-27, 2005
    8. Vortex pinning for the Ginzburg-Landau equations with normal impurity inclusion,
        The Fourth Nonlinear Congress at Orland, USA, July, 2004
    9. Multiple stable patterns for some reaction-diffusion equation in disrupted environments,
        Conference on Dynamical Systems and Differential Equations at Pomona, USA, June, 2004
    10. Symmetry breaking phenomena in an optimaization problem for some nonlinear elliptic equation,
        "International Conference on Nonlinear Partial Differential Equations", at City University of Hong Kong, September, 2002
    11. Eigenvalue Optimaizations and the Logistic Equation with Diffusion
        12. Symposium on "Professor K.Mochizuki's retirement" at Tokyo Metropolitan University, Feb. 2002
    12. Existence of positive solutions for some nonlinear elliptic equation on unbounded domains with cylindrical ends
        13. Heat Equations and related topics at Tohoku University, Oct. 2001
    13. A Minimization problem for the Frist Dirichlet eigenvalue and numerical results
        14. RIMS "Potential Theory and Related Topics", June 1999
    14. Optimization problems for Dirichlet eigenvalues to elliptic operators
        15. New trends in Nonlinear Partial Differential Equations at Ryukoku University, June 1999
    15. On a one-dimensional variational problem related to Cahn-Hilliard energy in a strip-like domain
        16. Workshop on Reaction-Diffusion equations,
          Hong Kong University, March 1999

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