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dirichlet problem for the laplacian in the strip

pantboio

Member
Nov 20, 2012
45
I'm looking for all functions $u$ harmonic in $S$ and continous in $\overline S$ such that
$$u(a,y)=u(b,y)=0,\forall y$$
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?
 

chisigma

Well-known member
Feb 13, 2012
1,704
I'm looking for all functions $u$ harmonic in $S$ and continous in $\overline S$ such that
$$u(a,y)=u(b,y)=0,\forall y$$
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?
As preliminary inspection we can follow the classical approach assuming that is $\displaystyle u(x,y)= \alpha(x)\ \beta(y)$ and that leads to the pair of ODE...


$\displaystyle \alpha^{\ ''} + \lambda \alpha =0$ (1)

$\displaystyle \beta^{\ ''} - \lambda \beta =0$ (2)

... where $\lambda$ is a constant that will be better defined later.

For semplicity sake we suppose that $0 \le x \le 1$. Starting from (1) its [not identically equal to zero...] solution with the contour conditions $\displaystyle \alpha(0)= \alpha(1)=0$ is...

$\displaystyle \alpha= k\ \sin \sqrt{\lambda}\ x = k\ \sin \pi\ n\ x $ (3)

... where k is a constant and $\sqrt{\lambda}= \pi\ n$.

Now we observe (2), the solution of which is...

$\displaystyle \beta = c_1\ e^{\sqrt {\lambda}\ y} + c_{2}\ e^{- \sqrt{\lambda}\ y}$ (4)

... and soon a little problem appears. If the condition is $\displaystyle \lim_{|y| \rightarrow \infty} u(x,y)=0$ and S is defined in $0 \le x \le 1$ and $- \infty < y < + \infty$, then in (4) is $c_{1}=c_{2}=0$ and the Diriclet problem has the only solution is $u(x,y)=0$. If S, for example, is defined in $0 \le x \le 1$ and $0 \le y < + \infty$, then the solution is...

$\displaystyle u(x,y)= \sum_{n=1}^{\infty} k_{n}\ \sin (\pi\ n\ x)\ e^{- \pi\ n\ y}$ (5)

Kind regards

$\chi$ $\sigma$
 
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