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Problem set 7, PDE

Problem 1

  1. If $latex u$ is harmonic in the connected domain $latex \Omega$ and is not constant, then $latex u(\Omega)$ is open in $latex \R$.

  2. Suppose $latex \Omega$ is bounded and that its boundary $latex \partial\Omega$ is connected. If $latex u$ is harmonic in $latex \Omega$, then $latex u(\Omega)\subset u(\partial\Omega)$.

Problem 2

  1. A radial harmonic function on $latex \mathbb B$ is constant.

  2. A positive harmonic function on $latex \R^d$ is constant.

Problem 3

Suppose $latex u(x)$ is harmonic in some domain in $latex \R^n$. Then

$latex v(x) = |x|^{2-n} u\Big(\dfrac{x}{|x|^2}\Big)$

is also harmonic in a suitable domain.

Problem 4

For $latex n=2$, find the Green's function for the Laplace operator on the first quadrant.