Any harmonic function $u$ on a simply connected domain in $\mathbb{R}^2$ is the real part of a holomorphic function. If the domain is multiply connected, then this is no longer true: the harmonic conjugate of $u$ may have periods.


I wonder if the following is true: let $C\_1, \ldots, C\_n$ be the components of the boundary of a domain, and let functions $f\_i \colon C\_i \to \mathbb{R}$ be given. Do there exist constants $a\_1, \ldots, a\_n$ such that the solution to the Dirichlet problem $\Delta u = 0$, $u \mid\_{C\_i} = f\_i + a\_i$ is the real part of a holomorphic function?


This is equivalent to the non-degeneracy of a matrix (periods of the conjugates of the solutions to several Dirichlet problems, the $i$-th problem being $f\_j = \delta\_{ij}$), and maybe follows from some classical facts about Riemann surfaces, but I am a stranger to the field...



Periods of the harmonic conjugate and a Dirichlet problem on a multiply connected domain
The answer is yes and you can find it in the book [1], chapter 1, §4, theorem 4.3, pp. 20-22. Precisely Wen, by constructing a suitable harmonic function and its harmonic conjugate, proves that on a $(N+1) $-connected domain in $\Bbb C$ whose connected components of the boundary $\Gamma$ are $C\_0,\ldots,C\_N$, there exists a unique analytic function $F(z)$ such that
$$
\begin{cases}
\Re F(t) =\Phi(t) + h(t) & t\in\Gamma=\bigcup\_{i=0}^NC\_i\\
\Im F(a) = b & a\in C\_0
\end{cases}
$$
where


* $\Phi(t)\in C^{1,\alpha}(\Gamma)$ is a real valued function on the boundary of $D$ (the requirement can be relaxed in $\Phi(t)\in C^{0,\alpha}(\Gamma)$ by a suitable approximation as explained in [1], p. 20 footnote 2)
* The function $h(t)$ is a simple function defined as
$$
h(t)=
\begin{cases}
0, & t\in C\_0 \\
h\_j & t\in C\_j, \; j= 1,\ldots, N.
\end{cases}
$$
where the $\{h\_j\}\_{j=1,\ldots,N}$ are indeterminate real constants
* $b$ is a real constant.


**Reference**


[1] Guo-Chun Wen, *Conformal mappings and boundary value problems*, translated from the Chinese by Kuniko Weltin (English), Translations of Mathematical Monographs 106, Providence, RI: American Mathematical Society (AMS). viii, 303 p. (1992), ISBN 0-8218-4562-4, [MR1187758](http://www.ams.org/mathscinet-getitem?mr=MR1187758), [Zbl 0778.30011](https://zbmath.org/?q=an%3A0778.30011)


