Nonlinear Neumann Problems for Fully Nonlinear Elliptic PDEs on a Quadrant

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 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 (Hi in the main text) describing the boundary condition are required to be positively onehomogeneous, and the result in this paper removes the positive homogeneity in two-dimension. An existence result for solutions is also presented.