A wire may be thought of as a smooth compact curve $C \subset \mathbb{R}^3$ with boundary two endpoints. Suppose we are given a smooth $\phi: C \to \mathbb{R}$ (a potential on the wire), then can $\phi$ be uniquely extended to $\mathbb{R}^3$ such that $$\triangle \phi = 0 \quad \text{on } \mathbb{R}^3 \setminus C $$ and $\phi = o(1)$ at $\infty$ ?
Is there an explicit solution for interesting curves, such as a circular arc with potentials (somehow) increasing from one endpoint to the next?
This would model the field outside a wire.
I don't think there is a unique answer to this question.
Consider for example a circle $C$ on the xy plane and two spheres $S_1$ and $S_2$ kept at constant potential $V_0$ that cross each other sharing as their common boundary the circle C. It is obvious that the circle is kept at the same constant potential in both cases, but the electric fields produced by these two configurations are different if $V_0\neq V(r=\infty)$, since the two spheres are necessarily centered at different points in $\mathbb{R}^3.$