Calc. Var. (2010) 37:485–522 DOI 10.1007/s00526-009-0274-x Calculus of Variations A class of integral equations and approximation of p-Laplace equations Hitoshi Ishii · Gou Nakamura Received: 18 May 2009 / Accepted: 20 September 2009 / Published online: 13 October 2009 © Springer-Verlag 2009 Abstract Let  ⊂ 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  with the boundary condition u = g 0 on ∂, where f 0 ∈ C ( ) and g 0 ∈ C (∂) are given functions and M is the singular integral operator given by M[u ](x ) = p.v.  B(0,ρ(x )) p - σ |z | n+σ |u (x + z ) - u (x )| p-2 (u (x + z ) - u (x )) dz , with some choice of ρ ∈ C ( ) having the property, 0 < ρ(x ) ≤ dist (x ,∂). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on , 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 ν p u = f 0 in  with the Dirichlet condition u = g 0 on ∂, where the factor ν is a positive constant (see (7.2)). Mathematics Subject Classification (2000) Primary 45G05 · 35J60 · 45M05 Dedicated to Professor Luis A. Caffarelli on the occasion of his 60th birthday. H. Ishii (B ) Department of Mathematics, Waseda University, Nishi-Waseda, Shinjuku, Tokyo 169-8050, Japan e-mail: hitoshi.ishii@waseda.jp G. Nakamura Department of Pure and Applied Mathematics, Waseda University, Ohkubo, Shinjuku, Tokyo 168-8555, Japan e-mail: g-nakamura@fuji.waseda.jp 123