Nonlinear Neumann Problems for Fully Nonlinear Elliptic PDEs on a Quadrant

Abstract

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 et al. [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 dimensions. An existence result for solutions is also presented.