In this paper, we establish some estimates to demonstrate the smoothness of solution for the Neumann boundary problem. To achieve this, we employ the Calderón-Zygmund theory for nonlinear PDEs. We consider bounded non-smooth domains and derive Hölder estimates for solutions of the Neumann problem associated with nonlinear elliptic equations. These estimates are crucial for understanding the regularity of solutions, which is essential for their application in practical problems. Specifically, the smoothness results obtained in this paper are directly applicable in engineering management, data analysis, and oil and gas filtration processes, where accurate modeling of fluid flow, stress distribution, and heat transfer in irregular domains is required.