Problem set 3, Harmonic Analysis

Due February 22

Problem 1: Symmetry Lemma

If $latex x\in\mathbb B$ and $latex \xi\in\mathbb S$, then

$latex \displaystyle \Big| |x|\xi - \frac{x}{|x|} \Big| = |x - \xi|.$

Problem 2: Hopf Lemma

If $latex u$ is a nonconstant harmonic function in $latex \mathbb B$, is continuous on $latex \bar{\mathbb B}$, and attains its maximum at $latex \zeta\in\mathbb S$, then there exists $latex c > 0$ such that

$latex u(\zeta) - u(r\zeta) > c (1 - r)$

for any $latex 0 < r < 1$.

Problem 3

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

Problem 4

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

Problem 5

Let $latex u$ be harmonic with an isolated singularity at $latex x_0$. If

$latex \displaystyle \lim_{x\to x_0} |x-x_0|^{d-2} u(x) = 0,$

then $latex x_0$ is a removable singularity.