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## Due February 22

### Problem 1: Symmetry Lemma

If $x\in\mathbb B$ and $\xi\in\mathbb S$, then
$\displaystyle \Big| |x|\xi - \frac{x}{|x|} \Big| = |x - \xi|.$

### Problem 2: Hopf Lemma

If $u$ is a nonconstant harmonic function in $\mathbb B$, is continuous on $\bar{\mathbb B}$, and attains its maximum at $\zeta\in\mathbb S$, then there exists $c > 0$ such that
$u(\zeta) - u(r\zeta) > c (1 - r)$
for any $0 < r < 1$.

### Problem 3

If $u$ is harmonic in $\Omega$ and $\bar B_r(x_0)\subset\Omega$, then the values of $u$ in $B_r(x_0)$ are determined by its values on $S_r(x_0)$.

### Problem 4

Let $u_n$ be a sequence of harmonic functions in $\Omega$ such that $u_n\rightrightarrows u$ on any compact $K\subset\Omega$. Then $u$ is harmonic in $\Omega$.

### Problem 5

Let $u$ be harmonic with an isolated singularity at $x_0$. If
$\displaystyle \lim_{x\to x_0} |x-x_0|^{d-2} u(x) = 0,$
then $x_0$ is a removable singularity.