Given a bounded smooth domain Ω of R 2 satisfying the exterior circle condition with radius r and a smooth boundary data φ on ∂Ω, we prove that if r is bigger than a constant (explicitly calculated) depending only on the C 2 norm of φ then the Dirichlet problem for the minimal surface equation for Ω and φ has a solution. Since the condition on r is trivially satisfied if the domain is convex, our result generalizes the classical theorem of R. Finn [F].

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