This is a Laplace equation on an annulus $1\leq x^2+y^2 \leq 4$, where $u(2,\theta)=f(\theta)$ and $u(1,\theta)=g(\theta)$. I have never understood how we determine boundary equations, which ones to use, and how many we need. One has to do with $f(\theta)=f(2\pi +\theta)$, right?
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The boundary conditions that are typically chosen in this scenario are motivated physically. Since the solution to Laplace's equation is the electrostatic potential in the context of electromagnetism, one would expect periodicity in the potential and the magnitude of the electric field. Assuming to the contrary would lead to things that haven't been observed in the lab, as far as I know. So to add to your requirement, $$f_{\theta}=f_\theta(2\pi+\theta)\quad\forall\theta.$$ In addition, if you are solving Laplace's equation inside of the disk, then $$\lim_{r\to 0}|u(r,\theta)|<\infty,$$ or outside $$\lim_{r\to \infty}|u(r,\theta)|<\infty.$$ This requires that the solution exist everywhere in your domain of definition. Since you are solving your problem on an annular region, this doesn't apply now, but helps in that standard circumstance. Luckily, you've completely specified the potential along the two enclosing boundaries. As a general rule of thumb, you need 1 boundary/initial condition for each derivative, so you have the four that you need.
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