Abstract
Let \({\Omega\subset\mathbb{R}^n}\) be a bounded domain, and let 1 < p < ∞ and σ < p. We study the nonlinear singular integral equation
with the boundary condition u = g 0 on ∂Ω, where \({f_0\in C(\overline\Omega)}\) and \({g_0\in C(\partial\Omega)}\) are given functions and M is the singular integral operator given by
with some choice of \({\rho\in C(\overline\Omega)}\) having the property, 0 < ρ(x) ≤ dist (x, ∂Ω). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on \({\overline\Omega}\), 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)).
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Dedicated to Professor Luis A. Caffarelli on the occasion of his 60th birthday.
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Ishii, H., Nakamura, G. A class of integral equations and approximation of p-Laplace equations. Calc. Var. 37, 485–522 (2010). https://doi.org/10.1007/s00526-009-0274-x
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DOI: https://doi.org/10.1007/s00526-009-0274-x