This paper characterizes the solution to the second order divergence form elliptic equation with mixed boundary condition on the upper half-space. By using the reflection method from Strauss's book [PDEs. An introduction, 2ed, 2008], we derive that the solution (Neumann harmonic function) has an integral representation of the Dirichlet boundary value. As applications, the real-variable characterizations of Neumann harmonic functions with Lebesuge, Hardy and BMO traces are established, respectively, via the gluing technology. Finally, the boundary value problem for the parabolic equation is also considered.

All authors thank Professor Jun Cao for many helpful discussions during the preparation of this work. We would like to thank the referee for very helpful comments and corrections. Jiahe Chen is supported by the National Innovation and Entrepreneurship Training Program for College Students (202410354030). Bo Li is supported by the NNSF of China (12201250), Zhejiang NSF of China (LQ23A010007), Jiaxing NSF of China (2023AY40003), and the Qin Shen Scholar Program of Jiaxing University. Bolin Ma is supported by the NNSF of China (11871452). Chao Zhang is supported by the NNSF of China (11971431) and Zhejiang NSF of China (LY22A010011).