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## Problem 1

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

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

## Problem 2

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

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

## Problem 3

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

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

is also harmonic in a suitable domain.

## Problem 4

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