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If L_0 and L_1 are two operators on a Lipschitz domain such that the L^p Dirichlet problem for the operator L_0 is solvable for some p in (1,\\infty) and the coefficients of the two operators are sufficiently close in the sense of Carleson measure, then the L^p Dirichlet problem for the operator L_1 is solvable for the same p. This is an improvement of the A_{\\infty} version of this result proved by Rios in \"The L^p Diriclet problem and nondivergence harmonic measure\" (Trans. AMS 355, 2 (2003)). 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