Problem set 2, Harmonic Analysis

Due February 15

Problem 1

Suppose $latex u$ is harmonic in a neighborhood of $latex \bar\Omega$, where $latex \Omega$ is a $latex C^1$ domain. Then

$latex \displaystyle \int_{\partial\Omega} \partial_\nu u \, d\sigma = 0$.

Problem 2

If $latex \Omega\subset\mathbb R^d$ is a bounded domain and $latex u$ is harmonic in $latex \Omega$ and continuous on $latex \bar\Omega$, then $latex u$ takes its maximum and its minimum on $latex \partial\Omega$.

Problem 3

Let $latex \Omega\subset\mathbb R^d$ be a bounded domain, $latex u$ and $latex v$ harmonic in $latex \Omega$ and continuous on $latex \bar\Omega$. If $latex u=v$ on $latex \partial\Omega$, then $latex u=v$ in $latex \Omega$.

Problem 4

If $latex f$ is an entire function and its real part is nonnegative, then $latex f$ is constant.

Problem 5

If $latex u$ is a radial harmonic function in $latex \mathbb B$, then it is constant.