  We consider the nonlinear Neumann problem for fully nonlinear elliptic PDEs
on a quadrant. We establish a comparison theorem for viscosity sub and
supersolutions of the nonlinear Neumann problem. The crucial argument in the
proof of the comparison theorem is to build a $C^{1,1}$ test function which
takes care of the nonlinear Neumann boundary condition. A similar problem has
been treated on a general $n$-dimensional orthant by Biswas, Ishii, Subhamay,
and Wang [SIAM J. Control Optim. 55 (2017), pp. 365--396], where the functions
($H_i$ in the main text) describing the boundary condition are required to be
positively one-homogeneous, and the result in this paper removes the positive
homogeneity in two-dimension. An existence result for solutions is also
presented.
