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Consider $\Omega:=]0,\pi[\times ]0,\infty[$. Use the Fourier method of separation to determine to (formal) bounded solution of the following task:

$\Delta u=0$ in $\Omega$

$u(0,y)=u(\pi,y)=0$ for $y\geq 0$

$u(x,0)=g(x)$ fpor $x\in [0,\pi]$,

whereat $g\in C^{0,\lambda}([0,\pi])$ with $0<\lambda\leq 1$ and $g(0)=g(\pi)=0$.

Yesterday my task was to https://math.stackexchange.com/questions/572293/find-the-bounded-solution-by-separation and I found it: $$ u(x,y)=\sum_{k=1}^{\infty}g_k\exp(-ky)\sqrt{\frac{2}{\pi}}\sin(kx), g_k:=\int_{0}^{\pi}g(x)\sqrt{\frac{2}{\pi}}\sin(kx)\, dx $$

Now I have to deal with two tasks that tie in with that solution.

(1) Show that the found solution is in $C^{\infty}(\Omega)\cap C(\overline{\Omega})$.

(2) Find the solution for $g(x):=2\sin(x)+5\sin(3x)$.

Concerning (1), I do not have an idea how to show that.

Concerning (2), my result is $$ u(x,y)=\frac{2}{\pi}\sum_{k=1}^{\infty}\sin(k\pi)\sin(kx)\exp(-ky)\cdot\left(\frac{1}{k+1}-\frac{1}{k-1}+\frac{5}{2}\left(\frac{1}{k+3}-\frac{1}{k-3}\right)\right). $$

Would be great to get a help for (1) and a feedback for (2).

Edit concerning (1)

In order to show that $u\in C(\overline{\Omega})$, I want to show that $$ \sum_{k=1}^{\infty}g_k\exp(-ky)\sqrt{\frac{2}{\pi}}\sin(kx) $$ converges uniformly, using Weierstraß.

For all $0<y<\infty$ it is $$ \lvert g_k\exp(-ky)\sqrt{\frac{2}{\pi}}\sin(kx)\rvert <M\lvert \exp(-ky)\rvert, $$ because $\lvert g_k\rvert\leq\lvert\int_0^{\pi}\vert g(x)\rvert <\infty$, because $g$ is Hölder-continious on $[0,\pi]$ and therefore integrable over $[0,\pi]$. Furthermore $\lvert\sin(kx)\rvert\leq 1$.

Additionally it is $\lvert\exp(-ky)\rvert=\exp(-ky)$ for all $0<y<\infty$ and all $k\geq 1$.

It is

$$ \sum_{k=1}^{\infty}\exp(-ky)<\infty. $$ So with Weierstraß the series above converges uniformly. Because all the $$ g_k\exp(-ky)\sqrt{\frac{2}{\pi}}\sin(kx) $$ are continious on $C(\overline{\Omega})$, $u$ is continious there.

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    The first thing I'd try is to differentiate $u(x,y)$ and see what you get. Key thing here is that no matter how many times you differentiate, you will always get a factor of $\exp(-ky)$ out front. What can you say about $$\sum_{k=1}^\infty e^{-ky}\cdot(\text{bounded junk})?$$ – snar Nov 19 '13 at 15:32
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    For $u\in C(\overline\Omega)$, it seems useful if you can prove that the series converges uniformly. For $u\in C^\infty(\Omega)$, you only need uniform convergence of the series for any derivative on compact subsets of $\Omega$ (in particular, $y\ge c$ for some positive constant $c$ in such a subset). – Harald Hanche-Olsen Nov 19 '13 at 15:40
  • Concerning $u\in C(\overline{\Omega})$ I tried to follow your hint (see edit in my question above). // But I do not understand your second hint concerning $u\in C^{\infty}(\Omega)$. Could you please explain why this is enough to show? –  Nov 19 '13 at 19:48
  • Did you forget me? –  Nov 21 '13 at 08:51

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