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We consider viscosity solutions of fully nonlinear elliptic equations for which u ≡ 0 is a solution. We assume F is smooth and uniformly elliptic only in a neighborhood of the points (0, 0, 0, x), and show that u is smooth in the interior if ‖u‖ L ∞ is sufficiently small.
Ovidiu Savin (2007) studied this question.