### Abstract

Let Ω ⊂ ℝ^{n} be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation M[u](x) = f_{0}(x) in Ω with the boundary condition u = g_{0} on ∂Ω, where f_{0} ε C(Ω̄) and g_{0} 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 u_{σ} of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation νΔ_{p}u = f_{0} in Ω with the Dirichlet condition u = g_{0} on ∂Ω, where the factor ν is a positive constant (see (7.2)).

Original language | English |
---|---|

Pages (from-to) | 485-522 |

Number of pages | 38 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 37 |

Issue number | 3-4 |

DOIs | |

Publication status | Published - 2010 |

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### ASJC Scopus subject areas

- Analysis
- Applied Mathematics

### Cite this

*Calculus of Variations and Partial Differential Equations*,

*37*(3-4), 485-522. https://doi.org/10.1007/s00526-009-0274-x

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

Research output: Contribution to journal › Article

*Calculus of Variations and Partial Differential Equations*, vol. 37, no. 3-4, pp. 485-522. https://doi.org/10.1007/s00526-009-0274-x

}

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 -