By the maximum principle, every harmonic function on a bounded domain is uniquely determined by its boundary values. However, for unbounded domains, we can have infinitely many harmonic functions with prescribed boundary values. My question, roughly, is this: since we may use conformal maps between bounded and unbounded domains to construct harmonic functions, how do we still maintain uniqueness?
More concretely, suppose we have a harmonic function $F$ on the upper half plane with prescribed boundary values. Consider the conformal map $u(z)=i(1-z)/(1+z)$ that takes the disc to the upper half plane. Then $F \circ u + cy$ for any constant $c$ gives an infnitely family of harmonic functions on the disc with given boundary values. But this contradicts the remark above that harmonic functions on bounded domains are unique. Where is the flaw in the reasoning?