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We study uniformly elliptic fully nonlinear equations $$ F(D^2u, Du, u, x)=0, $$ and prove results of Gidas--Ni--Nirenberg type for positive viscosity solutions of such equations. We show that symmetries of the equation and the domain are reflected by the solution, both in bounded and unbounded domains.

Symmetry results for viscosity solutions of fully nonlinear uniformly elliptic equations