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

作者：Ishii Hitoshi^*; Nakamura Gou

来源：Calculus of Variations and Partial Differential Equations, 2010, 37(3-4): 485-522.

DOI：10.1007/s00526-009-0274-x

Let Omega subset of R(n) be a bounded domain, and let 1 < p < 8 and sigma < p. We study the nonlinear singular integral equation
M[u](x) = f(0)(x) in Omega
with the boundary condition u = g(0) on partial derivative Omega, where f(0) is an element of C(<(Omega)over bar>) and g(0) is an element of C(partial derivative Omega) are given functions and M is the singular integral operator given by
M[u](x) = p. v. integral(B(0, rho(x))) p - sigma/vertical bar z vertical bar(n+sigma) vertical bar u(x + z) - u(x)vertical bar(p) (2)(u(x + z) - u(x)) dz,
with some choice of rho is an element of C(Omega) having the property, 0 < rho(x) <= dist (x, partial derivative Omega). We establish the solvability (well-posedness) of this Dirichlet problem and the convergence uniform on <(Omega)over bar> as sigma -> p, of the solution us of the Dirichlet problem to the solution u of the Dirichlet problem for the p-Laplace equation nu Delta(p)u = f(0) in Omega with the Dirichlet condition u = g(0) on partial derivative Omega, where the factor nu is a positive constant (see (7.2)).

• 出版日期2010-3