Some exercises in general physics ask for students to find specific charge distributions on the boundaries of given conductors in various of situations. Usual answers goes like follows; firstly using symmetries or other techniques, e.g. image charges, to 'guess' how the distribution looks like, and then completes solution testing the boundary conditions, vanishing electric fields, etc. Almost surely, a mathematically oriented one may suspect that other possibilities maybe possible. So this question is almost surely duplicated. So let me state precise questions in the following form.
Question 0 Is question 1 duplicates or nonsense? If not, ignore question 0. If so please indicate the places where this question appeared or the reason it is trivially wrong or true.
Question 1 Is the following statement is true: Let $V_e: \mathbb{R}^3 \longrightarrow \mathbb{R}$ be a smooth function which vanishes at infinity uniformly. Let $C$ be a finite union of a collection of pairwise disjoint bounded domains say $C_1$, $C_2$, $\ldots$, $C_n$ in $\mathbb{R}^3$ whose boundary $S$ is a union of smooth surfaces $S_1$, $S_2$, $\ldots$, $S_n$ which are mutually non-overlapping(that is for any $i \neq j$, $S_i \cap S_j$ has no interior in $S_i$ with respect to the induced topology on $S_i$). For each $j=1,2, \ldots , n$, let $Q_j$ be a real number which represents the total charge on $S_j$. Then there is unique surface distribution $\sigma$ on $S$ so that the potential $$V(y) = \int_S \frac{\sigma(x)}{\hbox{dist}(x,y)} dx + V_e(y) $$ is constant on each of connected component of $C$ where $\int_{S_j} \sigma (x) dx =Q_j$ for each $j=1,2, \ldots, n$