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Let $\Omega\subset\mathbb R^n$ open, not bounded and $n\ge3$. Let $\partial\Omega$ bounded and regular concering the laplace operator. Given a continuous function $\phi:\partial\Omega\rightarrow\mathbb R$ and $\gamma\in\mathbb R$ there exists a harmonic function $u\in C^2(\Omega)\cap C^0(\overline\Omega)$ with $u=g\space\space\text{on}\space\partial\Omega$ and $\lim\limits_{|x|\rightarrow\infty}u(x)=\gamma$

How can you prove the existence?

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up vote 2 down vote accepted

This is very nontrivial and called the Dirichlet problem.

If you are looking for a solution using complex analysis (well, methods connected to complex analysis), both Ahlfors and Gamelin's textbooks on complex analysis give a proof of existence using subharmonic functions. This is called "Perron's method." I have looked at this closely (I was tested over it) and it gives good insight into exactly how pathological the boundary can be before you can no longer solve the problem.

Another route to existence is given the third volume of Stein and Shakarchi's four volume Princeton Lectures in Analysis series, in the book titled Real Analysis. I have not read it closely, but it might be a useful reference if you are not familiar with complex analysis or subharmonic functions.

You question is a little confusing to me because you say that $\Omega$ is not bounded, but $\partial\Omega$ is. Both references I gave only deal with the case where the region is bounded. However, I have no doubt that armed with the magic search term "Dirichlet Problem," you will be able to find literature about the unbounded case. This is a classic problem, and I'm sure others can give more authoritative references if you are looking for something research-level.

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Complex analysis is not very helpful here because the existence result stated by @Niro is false for $n=2$. –  user31373 Jun 15 '12 at 1:34
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You are right, I read the question too quickly. I will leave this answer up, as it is the only one here, but perhaps you could write something a bit more on-point? –  Potato Jun 15 '12 at 1:38
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