Additional Problems for Chapter 7 \( \newcommand{\ds}{\displaystyle} \newcommand{\diam}{\operatorname{diam}} \newcommand{\area}{\operatorname{area}} \newcommand{\dist}{\operatorname{dist}} \newcommand{\ord}{\operatorname{ord}} \newcommand{\res}{\operatorname{res}} \newcommand{\wind}{\operatorname{W}} \newcommand{\Aut}{\operatorname{Aut}} \newcommand{\SL}{\operatorname{SL}} \newcommand{\PSL}{\operatorname{PSL}} \newcommand{\iso}{\stackrel{\cong}{\longrightarrow}} \newcommand{\myint}{\operatorname{int}} \newcommand{\ve}{\varepsilon} \newcommand{\es}{\emptyset} \newcommand{\sm}{\smallsetminus} \newcommand{\bd}{\partial} \newcommand{\Chat}{\hat{\mathbb C}} \newcommand{\Cstar}{{\mathbb C}^*} \newcommand{\Dstar}{{\mathbb D}^*} \newcommand{\myre}{\operatorname{Re}} \newcommand{\myim}{\operatorname{Im}} \newcommand{\ov}{\overline} \newcommand{\io}{\iota} \newcommand{\con}{\operatorname{const.}} \newcommand{\OO}{{\mathcal O}} \newcommand{\MM}{{\mathcal M}} \newcommand{\FF}{{\mathcal F}} \newcommand{\CC}{{\mathbb C}} \newcommand{\PP}{{\mathbb P}} \newcommand{\RR}{{\mathbb R}} \newcommand{\HH}{{\mathbb H}} \newcommand{\TT}{{\mathbb T}} \newcommand{\II}{{\mathbb I}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\DD}{{\mathbb D}} \newcommand{\QQ}{{\mathbb Q}} \)




Additional Problems for Chapter 7

  1. Let $U \subset \CC$ be a simply connected domain and $h: U \to \RR$ be a non-vanishing harmonic function. Show that there are harmonic functions $u,v: U \to \RR$ such that $h=u^2-v^2$.
  2. Let $h$ be a positive harmonic function in $\DD$. Show that there are harmonic functions $u,v: \DD \to \RR$ such that $h=e^u \sin v$.
  3. Let $u : \CC \to \RR$ be a harmonic function which satisfies \[ u(z) \leq a |\log |z|| + b \] for some constants $a,b > 0$. Prove that $u$ is constant.
  4. Suppose $U \subset \CC$ is a domain and $u,v: U \to \RR$ are harmonic.

    (i) Show that $u^2$ is harmonic in $U$ if and only if $u$ is constant.

    (ii) Show that $uv$ is harmonic in $U$ if and only if $u+icv$ is holomorphic in $U$ for some $c \in \RR$.

  5. Suppose $h$ is harmonic in $\CC$ and $\iint_\CC |h|^p \, dx dy <+\infty$ for some $p>1$. Show that $h$ must be identically zero. (Hint: Use the mean value property.)
  6. Suppose $u :\DD \to \RR$ is a harmonic function with $|u| \leq 1$. Let $v: \DD \to \RR$ be the unique harmonic conjugate of $u$ which satisfies $v(0) = 0$. Prove that \[ |v(z)| \leq \frac{2}{\pi} \log \left( \frac{1+|z|}{1-|z|} \right) \ \ \text{for all} \ z \in \DD. \] (Hint: Use Corollary 7.23.)
  7. Let $0<\delta<1$. Solve the Dirichlet problem $\Delta u=0$ in the domain $\{ z : \myre(z) > 0, |z-1| > \delta \}$, subject to the boundary conditions $u=0$ on the line $\myre(z)=0$ and $u=1$ on the circle $|z-1|=\delta$. Analyze what happens to the solution as $\delta \to 1$.
  8. If $f: \DD \to \Dstar$ is holomorphic, prove that \[ |f(z)| \leq |f(0)|^{\tfrac{1-|z|}{1+|z|}} \qquad \text{for all} \ z \in \DD. \]
  9. Suppose $U \subset \CC$ is a domain, $p \in U$, and $f_n=u_n+iv_n:U \to \CC$ are holomorphic with $f_n(p)=0$. If $u_1 \leq u_2 \leq u_3 \leq \cdots$, show that $\{ f_n \}$ converges compactly in $U$.

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