A class of integral equations and approximation of p-Laplace equations

Hitoshi Ishii, Gou Nakamura

    Research output: Contribution to journalArticle

    45 Citations (Scopus)

    Abstract

    Let Ω ⊂ ℝn be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f0(x) in Ω with the boundary condition u = g0 on ∂Ω, where f0 ε C(Ω̄) and g0 ε C(∂Ω) are given functions and M is the singular integral operator given by, with some choice of ρ ε C(Ω̄) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on Ω̄, as σ → p, of the solution uσ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔpu = f0 in Ω with the Dirichlet condition u = g0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).

    Original languageEnglish
    Pages (from-to)485-522
    Number of pages38
    JournalCalculus of Variations and Partial Differential Equations
    Volume37
    Issue number3-4
    DOIs
    Publication statusPublished - 2010

    Fingerprint

    P-Laplace Equation
    Laplace equation
    Dirichlet Problem
    Integral equations
    Integral Equations
    Approximation
    Mathematical operators
    Dirichlet conditions
    Singular Integral Operator
    Boundary conditions
    Singular Integral Equation
    Uniform convergence
    Well-posedness
    Solvability
    Bounded Domain
    Class

    ASJC Scopus subject areas

    • Analysis
    • Applied Mathematics

    Cite this

    A class of integral equations and approximation of p-Laplace equations. / Ishii, Hitoshi; Nakamura, Gou.

    In: Calculus of Variations and Partial Differential Equations, Vol. 37, No. 3-4, 2010, p. 485-522.

    Research output: Contribution to journalArticle

    @article{983b6ee9c0684c24a8ac9467cee9dc7f,
    title = "A class of integral equations and approximation of p-Laplace equations",
    abstract = "Let Ω ⊂ ℝn be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f0(x) in Ω with the boundary condition u = g0 on ∂Ω, where f0 ε C(Ω̄) and g0 ε C(∂Ω) are given functions and M is the singular integral operator given by, with some choice of ρ ε C(Ω̄) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on Ω̄, as σ → p, of the solution uσ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔpu = f0 in Ω with the Dirichlet condition u = g0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).",
    author = "Hitoshi Ishii and Gou Nakamura",
    year = "2010",
    doi = "10.1007/s00526-009-0274-x",
    language = "English",
    volume = "37",
    pages = "485--522",
    journal = "Calculus of Variations and Partial Differential Equations",
    issn = "0944-2669",
    publisher = "Springer New York",
    number = "3-4",

    }

    TY - JOUR

    T1 - A class of integral equations and approximation of p-Laplace equations

    AU - Ishii, Hitoshi

    AU - Nakamura, Gou

    PY - 2010

    Y1 - 2010

    N2 - Let Ω ⊂ ℝn be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f0(x) in Ω with the boundary condition u = g0 on ∂Ω, where f0 ε C(Ω̄) and g0 ε C(∂Ω) are given functions and M is the singular integral operator given by, with some choice of ρ ε C(Ω̄) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on Ω̄, as σ → p, of the solution uσ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔpu = f0 in Ω with the Dirichlet condition u = g0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).

    AB - Let Ω ⊂ ℝn be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f0(x) in Ω with the boundary condition u = g0 on ∂Ω, where f0 ε C(Ω̄) and g0 ε C(∂Ω) are given functions and M is the singular integral operator given by, with some choice of ρ ε C(Ω̄) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on Ω̄, as σ → p, of the solution uσ of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔpu = f0 in Ω with the Dirichlet condition u = g0 on ∂Ω, where the factor ν is a positive constant (see (7.2)).

    UR - http://www.scopus.com/inward/record.url?scp=77949774164&partnerID=8YFLogxK

    UR - http://www.scopus.com/inward/citedby.url?scp=77949774164&partnerID=8YFLogxK

    U2 - 10.1007/s00526-009-0274-x

    DO - 10.1007/s00526-009-0274-x

    M3 - Article

    AN - SCOPUS:77949774164

    VL - 37

    SP - 485

    EP - 522

    JO - Calculus of Variations and Partial Differential Equations

    JF - Calculus of Variations and Partial Differential Equations

    SN - 0944-2669

    IS - 3-4

    ER -