Let $p_0=0$, $p_1=1$, $p_2=1+i$ and $p_3=i$ be the four vertices of a square $Q$ on the complex plane $\mathbb C$.
Let $f \in C^{\infty}_c((0,1))$ and consider the following Riemann-Hilbert problem on $Q$:
$$ \frac{\partial}{\partial \bar{z}}F=0,\quad \text{on $Q$},$$
subject to
$$ \textrm{Re}\{F\}(t,0)=f(t) \quad \text{for $t \in (0,1)$}$$
and $$\textrm{Re}\{F\}_{|_{\partial Q\setminus (0,1)}}=0.$$
Does there exist a solution to this problem?
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3$\begingroup$ This does not look like a true Riemann–Hilbert problem: simply extend the boundary values to a harmonic function in the unit square to get the real part of $F$, and set the imaginary part of $F$ to be equal to the harmonic conjugate of the real part. Am I missing something? $\endgroup$– Mateusz KwaśnickiCommented Sep 17, 2020 at 18:53
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