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$$u(a,y)=u(b,y)=0,\forall y$$

and

$$\lim_{|y|\rightarrow +\infty} u(x,y)=0$$

where $S$ is the strip $\{a<\operatorname{Re}(z)<b\}$

My strategy is the following. I know that if $g$ is continous on $\partial D$, with $D$ the unit disc,then

\begin{equation*}

u(z)= \left\{

\begin{array}{}

\frac{1}{2\pi}\int_0^{2\pi}g(e^{i\theta}) \frac{1-|z|^2}{|e^{i\theta}-z|^2}d\theta& \text{if } z\in D,\\

g(z)& \text{if } z\in\partial D\\

\end{array} \right.

\end{equation*}

is harmonic in $D$, continous in $\overline D$ and $u=g$ on $\partial D$.

So i find a conformal map $\phi$ from the strip to the unit disc and i look for harmonic functions in the disc that vanish on the boundary. But in this case $g$ is the function identically equal to 0, hence the only harmonic functions i find is the zero one. In all this i have the strong sensation to have missed something fundamental from the theory, but i don't know what. Can someone give me a suggestion?