Given a space $\mathbb{R}^n$, can I say all functions $$u: \{x_1\geq 0\}\to \mathbb{R}$$ s.t. u is harmonic on $\{x_1\geq 0\}$, u is non-negative and u vanishes on the boundary $\{x_1=0\}$ is exactly $x=cx_1$ where c is a non-negative constant?
I do believe that it’s true because that we can use a sequence of harmonic polynomials to approach it (can we?) then derive that the polynomial is exactly $x_1$ up to a constant. But I can’t figure out the details. Moreover, I’m also wondering what kind of boundary conditions allows a non-negative harmonic function.