Construct a harmonic function satisfying given two conditions

Question

How to construct a harmonic function $u(x,y)$ such that $u(x,y)=7$ in $0<|z|\le 1$ and $u(x,y)=11$ in $5\le |z|<\infty$.

I am trying to define $u(x,y)$ in $1<|z|<5$ such that $u$ is twice differentiable satisfyning given two conditions and moreover $u$ is harmonic in that domai.

I don't know how to find out such function ! Any help?

Answer 1

No such function exist. Note that $u$ can be extended as $u(0)=7$ and $u(\infty)=11$, i.e. we have a harmonic function $u:\Bbb S^2\to \Bbb R$. Since, $\Bbb S^2$ is compact $u$ attatins its maximum at some point $p\in \Bbb S^2$. Let $z:V\to \Bbb C$ be a local chart at $p\in V$. Then, $u\circ z^{-1}:z(V)\to \Bbb R$ is harmonic and attains its maximum at some interior point, hence constant by maximum principle.

Thus the closed set $C\subseteq \Bbb S^2$ where $u$ attains its maximum is also open hence is all of $\Bbb S^2$.

In other words, on a compact Riemann surface every subharmonic or harmonic or holomorphic function is constant.