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Let $\Omega = \{x\in \mathbb R^n: |x|>1 \}$ and suppose $u \in C^2(\Omega) \cap C(\bar \Omega)$ is a bounded solution of the Dirichlet problem of Laplace equation: $$\Delta u = 0 ~\text{in}~ \Omega$$

and $u=\phi~ \text{on the boundary} ~\{|x|=1\}$, with $\phi$ continuous on the boundary.

We know when $n=2$, there is at most one solution of the above problem. Now my question is what about $n=3$?

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Consider the boundary value problem in $\mathbb{R}^n$: $\Delta u = 0$ and $u(x) = g(x)$ for $|x|=1$.

For any $n$, we can use Poisson's formula to get a harmonic function: $$ u(x) = \frac{1-|x|^2}{n\omega_n} \int_{\partial B} \frac{g(y)}{|x-y|^n} dS_y $$

Where $B$ is the unit ball ($|x| \leq 1$) in $\mathbb{R}^n$, and $\omega_n$ is the volume of $B$. It is relatively easy to show that $u \in C^2$ and is harmonic. It is slightly more difficult to prove that for any $y \in \partial B$ we have that $\lim_{x \to y} u(x) = g(y)$, satisfying the boundary condition.

To show that it is unique, generally we take advantage of a maximum principle. For example, it is known that non-constant harmonic functions on $B$ cannot attain their maximum or minimum on the interior of $B$. So, suppose that there are two solutions $u_1$ and $u_2$ to the boundary value problem. Then we have that $v(x)=u_1(x)-u_2(x)$ satisfies $\Delta v = 0$ and $v(x)=0$ for $x \in \partial B$. Since the maximum and minimum of $v$ must be attained on the boundary of $B$, we get that $0 \leq v \leq 0$ for all $x \in B$, which means that $v=0$, and thus $u_1=u_2$.

Proofs that $u$ satisfies the boundary condition, and the maximum principle generally tend to be nontrivial. They can be found in many upper level PDE books. My personal favorite is Renardy and Rogers' book Introduction to Partial Differential Equations.

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