I am trying to solve a linear pde of the form $\Delta u = f(u)$ using separation of variables in polar coordinates in the half-space $\mathbb{R}^2_{+}=\{(x,y):x\in \mathbb{R},y>0\}$, with the boundary condition $\frac{\partial}{\partial y} u(x,0)=0.$
In polar coordinates I know that $$\frac{\partial}{\partial y} u(r,\theta)=\frac{\partial u}{\partial r}\sin(\theta) + \frac{\partial u}{\partial \theta} \frac{\cos(\theta)}{r}$$ and that when $y=0$ then $r\sin(\theta)=0.$ My question is when $r\sin(\theta)=0$ which condition should I use $r=0$ (which does not make much sense) or $\theta =0,\pi?$ I am guessing that the boundary condition $\frac{\partial}{\partial y} u(x,0)=0$ will translate to the boundary condition $\frac{\partial}{\partial \theta} u(r,0)=\frac{\partial}{\partial \theta} u(r,\pi)=0$ in polar coordinates, right?