Ikoma, Norihisa

写真a

Affiliation

Faculty of Science and Technology, Department of Mathematics (Yagami)

Position

Associate Professor
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Papers 【 Display / hide 】

  • Existence and Regularity for Prescribed Lorentzian Mean Curvature Hypersurfaces, and the Born–Infeld Model

    Jaeyoung Byeon, Norihisa Ikoma, Andrea Malchiodi, Luciano Mari

    Annals of PDE (Springer Science and Business Media LLC)  10 ( 1 )  2024.06

    Accepted,  ISSN  25245317

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    Given a measure ρ on a domain Ω ⊂ Rm , we study spacelike graphs over Ω in Minkowski space with Lorentzian mean curvature ρ and Dirichlet boundary condition on ∂Ω , which solve [Figure not available: see fulltext.] The graph function also represents the electric potential generated by a charge ρ in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer uρ of the associated action Iρ(ψ)≐∫Ω(1-1-|Dψ|2)dx-⟨ρ,ψ⟩ among functions ψ satisfying | Dψ| ≤ 1 , by the lack of smoothness of the Lagrangian density for | Dψ| = 1 one cannot guarantee that uρ satisfies the Euler-Lagrange equation (BI). A chief difficulty comes from the possible presence of light segments in the graph of uρ . In this paper, we investigate the existence of a solution for general ρ . In particular, we give sufficient conditions to guarantee that uρ solves (BI) and enjoys log -improved energy and Wloc2,2 estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of ρ to ensure the solvability of (BI).

  • The existence and multiplicity of L^2-normalized solutions to nonlinear Schrödinger equations with variable coefficients

    Norihisa Ikoma, Mizuki Yamanobe

    Advanced Nonlinear Studies (Walter de Gruyter GmbH)  24 ( 2 ) 477 - 509 2024.04

    Accepted,  ISSN  1536-1365

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    Abstract

    The existence of L <sup>2</sup>–normalized solutions is studied for the equation − Δ u + μ u = f ( x , u )     in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R } }^{N},\quad {\int }_{ { \mathbf{R } }^{N } }{u}^{2} \mathrm{d}x=m.$ Here m &gt; 0 and f(x, s) are given, f(x, s) has the L <sup>2</sup>-subcritical growth and (μ, u) ∈ R × H <sup>1</sup>(R <sup> N </sup>) are unknown. In this paper, we employ the argument in Hirata and Tanaka (“Nonlinear scalar field equations with L <sup>2</sup> constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019) and find critical points of the Lagrangian function. To obtain critical points of the Lagrangian function, we use the Palais–Smale–Cerami condition instead of Condition (PSP) in Hirata and Tanaka (“Nonlinear scalar field equations with L <sup>2</sup> constraint: mountain pass and symmetric mountain pass approaches,” Adv. Nonlinear Stud., vol. 19, no. 2, pp. 263–290, 2019). We also prove the multiplicity result under the radial symmetry.

  • On weak solutions to a fractional Hardy–Hénon equation, Part II: Existence

    Shoichi Hasegawa, Norihisa Ikoma, Tatsuki Kawakami

    Nonlinear Analysis (Elsevier BV)  227   113165 - 113165 2023.02

    Accepted,  ISSN  0362-546X

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    This paper and Hasegawa et al. (2021) treat the existence and nonexistence of stable weak solutions to a fractional Hardy–Hénon equation (−Δ)su=|x|ℓ|u|p−1u in RN, where 0<s<1, ℓ>−2s, p>1, N≥1 and N>2s. In this paper, when p is critical or supercritical in the sense of the Joseph–Lundgren, we prove the existence of a family of positive radial stable solutions, which satisfies the separation property. We also show the multiple existence of the Joseph–Lundgren critical exponent for some ℓ∈(0,∞) and s∈(0,1), and this property does not hold in the case s=1.

  • Nonlinear elliptic equations of sublinearity: qualitative behavior of solutions

    Norihisa Ikoma, Kazunaga Tanaka, Zhi-Qiang Wang, Chengxiang Zhang

    Indiana University Mathematics Journal (Indiana University Mathematics Journal)  71 ( 5 ) 2001 - 2043 2022

    Accepted,  ISSN  0022-2518

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    We study existence, uniqueness, and qualitative property of solutions for a class of nonlinear elliptic equations of sublinearity. We also study the asymptotic behavior of a ground state solution for sublinear equations in RN and give estimates for a radius of its support set as various parameters involved, including the sublinearity power approaching to the limiting cases.

  • Existence and asymptotic behavior of positive solutions for a class of locally superlinear Schrödinger equation

    Adachi S., Ikoma N., Watanabe T.

    Manuscripta Mathematica (Manuscripta Mathematica)  172 ( 3-4 ) 933 - 970 2022

    ISSN  00252611

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    This paper treats the existence of positive solutions of - Δ u+ V(x) u= λ f(u) in RN. Here N≥ 1 , λ > 0 is a parameter and f(u) satisfies conditions only in a neighborhood of u= 0. We shall show the existence of positive solutions with potential of trapping type or G-symmetric potential where G⊂ O(N). Our results extend previous results (Adachi and Watanabe in J Math Anal Appl 507:125765, 2022; Costa and Wang in Proc Am Math Soc 133(3):787–794, 2005; do Ó et al. in J Math Anal Appl 342:432–445, 2008) as well as we also study the asymptotic behavior of a family (uλ)λ≥λ0 of positive solutions as λ → ∞.

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Papers, etc., Registered in KOARA 【 Display / hide 】

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Research Projects of Competitive Funds, etc. 【 Display / hide 】

  • 特異性や制約条件を持つ非線形楕円型方程式の解構造の研究

    2024.04
    -
    2028.03

    日本学術振興会, Grants-in-Aid for Scientific Research, Grant-in-Aid for Scientific Research (C), Principal investigator

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    本研究課題では，変分構造を持つ非線形楕円型方程式に対し，解の存在や解の性質を明らかにすることを目標にしている．本課題で扱う非線形楕円型方程式は物理学や幾何学において現れ，解の存在やその性質を調べることは重要であるが，解の陽的な表示を得ることは非常に難しく，ほとんど不可能であるように思える．そのため解の陽的な表示を使わずに特定の性質を持った解の存在を示したり，解があったとすると特定の性質を必ず持つ，ということを明らかにしたい．

  • Study of solution structures of elliptic equations: supercritical, critical and subcritocal cases

    2019.04
    -
    2024.03

    日本学術振興会, Grants-in-Aid for Scientific Research, Miyamoto Yasuhito, Grant-in-Aid for Scientific Research (B), No Setting

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    Solution structures, which are bifurcation diagrams, of supercritical elliptic Dirichlet problem are studied. It is known that standard variational approaches, which are used to study subcritical problems, are not applicable to supercritical problems. A large part of a solution structure was not known. In this study we consider positive solutions of elliptic Dirichlet problem when the domain is a ball. We show that a positive radial singular solution of the supercritical elliptic equation plays a crucial role in the study of bifurcation diagrams. We prove the existence and uniqueness of a positive radial singular solution and obtain an asymptotic expansion of the singular solution near the singular point. We apply our theory to several interesting elliptic equations and classify the bifurcation diagrams.

  • Study of the strucure of solutions to nonlinear elliptic equations with various effects

    2019.04
    -
    2023.03

    MEXT,JSPS, Grant-in-Aid for Scientific Research, Ikoma Norihisa, Grant-in-Aid for Scientific Research (C), Principal investigator

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    The aim of this research project was to investigate the structure of nontrivial solutions to elliptic equations involving singularities, nonlocalities and so on. During the period, the following results were obtained. For the Born-Infeld equation(this equation has a singularity), the regularity of minimizer as well as the relation between minimizers and weak solutions were studied. For the equation with the fractional Laplacian and the Hardy-Henon type nonlinearity (the equation has a nonlocality), the existence and nonexistence of stables solutions was proved. The layer property of the family of stable solutions were also shown. In addition to these two equations, the existence of nontrivial solutions and their properties were obtained for the equation with sublinear nonlinearities, a class of equations involving the 1 dimensional Pucci operators, the equation with large parameters and the equation with a constraint on the L^2 norm of solutions.

  • Analysis of large time behavior of solution to nonlinear partial differential equations with dispersion

    2017.04
    -
    2021.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Segata Jun-ichi, Grant-in-Aid for Scientific Research (B), No Setting

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    Nonlinear dispersive partial differential equation is one of important class in the partial differential equations. Due to a complex interaction between dispersive and nonlinear effects in the equation, there is a wide variety of asymptotic behavior of solution, and it is difficult to study long time behavior of solution. In this research, we tried to gain a new insight on long time behavior of solution to nonlinear dispersive equation by analyzing concrete models via harmonic analysis and variational methods.

  • Nonlinear elliptic partial differential equations having variation structure

    2016.04
    -
    2019.03

    Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, Ikoma Norihisa, Grant-in-Aid for Young Scientists (B), No Setting

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    In this project, the existence of solutions and their properties were studied for nonlinear elliptic partial differential equations. In particular, we treated equations which have variational structure (for instance, equations with fractional operators of elliptic operators). We proved the existence of solutions satisfying some properties and showed the existence of multiple solutions. We also studied a variant of the Trudinger-Moser inequality and found conditions when the inequality is satisfied as an equality. This inequality is related to a certain nonlinear elliptic partial differential equation.

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Courses Taught 【 Display / hide 】

  • TOPICS IN ANALYSIS

    2025

  • MATHEMATICS 3B

    2025

  • MATHEMATICS 3A

    2025

  • INDEPENDENT STUDY ON FUNDAMENTAL SCIENCE AND TECHNOLOGY

    2025

  • GRADUATE RESEARCH ON FUNDAMENTAL SCIENCE AND TECHNOLOGY 2

    2025

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