I have gotten stuck with a problem for PDEs class for a few days. I did not figure out how to start a solution for it.
Problem: Let $g \in C(\partial B(0, R))$, $n > 2$. Find a formula for a harmonic function $u : \mathbb{R}^n\setminus B(0, R) \to \mathbb{R}$ with $u(x) = g(x)$ on $|x| = R$, and $u(x) \to 0$ as $|x| \to \infty$. Is it unique? Is it still unique if we remove the condition $u(x) \to 0$ as $|x| \to \infty$?
I know how to find a formula for a harmonic function $u$ defined on an open bounded set, but I do not find any formula in Evans book can be applied to this situation. Plus, I have no idea how Green's functions are defined on unbounded set. Could you please help me to solve this problem?
Dan
If you know a general formula for a solution $u$ with $u=g$ on $|x|=R$, and $u(x) \rightarrow 0$ as $|x|\rightarrow\infty$, then replace $g$ by $g-1$ and solve for $u_{g-1}$ with these properties. Then $v=u_{g-1}+1$ is a solution of Laplace's equation with $v=g$ on $|x|=R$, and $v(x)\rightarrow 1$ as $|x|\rightarrow\infty$.