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)).
|ジャーナル||Calculus of Variations and Partial Differential Equations|
|出版ステータス||Published - 2010|
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