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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 1}
\date{due February 20, 2017}
\maketitle

\begin{enumerate}

\item Evans, p.\ 85 problem 2

\item Consider a function of one complex variable $w \colon \mathbb{C}
  \to \mathbb{C}$ on an open connected subset of the complex plane,
  and write $w=w(z)$ with $w=u+\mathrm{i} v$ and $z=x + \mathrm{i} y$.
  The function $w$ is called \emph{(complex) differentiable} or
  \emph{holomorphic} if $u$ and $v$ have continuous first partial
  derivatives with respect to $x$ and $y$ that satisfy the so-called
  \emph{Cauchy--Riemann equations}
\begin{gather*}
  u_x = v_y \\
  u_y = - v_x
\end{gather*}
It is known that holomorphic functions are infinitely differentiable.

Show that the real and imaginary parts of a holomorphic function are
harmonic.

\item Find all radial solutions for the modified Laplace equation
\[
  D \cdot (r\,Du(x)) = 0 \,,
\]
where $u \colon \R^n \to \R$ with $n \geq 2$, and $r = \lvert x
\rvert$.

\item Show that 
\begin{equation*}
  \Delta u = 2n \lim_{\varepsilon\to 0} 
  \frac{1}{\varepsilon^2} \, 
  \PVint_{\partial B(x,\varepsilon)} (u(y)-u(x)) \, dS(y) \,.
\end{equation*}

\item Evans, p. 85 problem 3.

\emph{Note:} This problem can be hard when you try it.  Please ask in
class in case you get stuck.


\end{enumerate}


\end{document}

