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| We prove that the existence of a solution to a fully nonlinear elliptic equation in a bounded domain $\Omega$ with an overdetermined boundary condition prescribing both Dirichlet and Neumann constant data forces the domain $\Omega$ to be a ball, under any of the following conditions: (a) the operator is $C^1$ in the second-order derivative, (b) the space dimension is 2, (c) the domain is strictly convex. This is a generalization of Serrin's classical result from 1971. |
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