Nonlinear Neumann problems for fully nonlinear elliptic PDEs on a quadrant

TL;DR

This paper proves a comparison theorem and establishes existence for nonlinear Neumann problems involving fully nonlinear elliptic PDEs on a quadrant, removing the positive homogeneity condition in two dimensions.

Contribution

It extends previous results by removing the positive homogeneity requirement for boundary functions in two-dimensional nonlinear Neumann problems.

Findings

Established a comparison theorem for viscosity solutions.

Constructed a $C^{1,1}$ test function for boundary conditions.

Proved existence of solutions under new conditions.

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, 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.