\documentclass[11pt]{article} 
\usepackage[T1]{fontenc}
\usepackage{amsmath}
\usepackage{amsxtra}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsthm}
\usepackage{enumitem}
\usepackage{verbatim}
\usepackage{url}
\usepackage{color,hyperref}
\hypersetup{colorlinks,breaklinks,
             linkcolor=blue,urlcolor=blue,
           anchorcolor=blue,citecolor=blue}

\usepackage[margin=3cm]{geometry}
\newcommand{\eps}{\varepsilon}
\newcommand{\Z}{\mathbb{Z}}
\renewcommand{\L}{\mathcal{L}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\x}{\mathbf{x}}
\newcommand{\n}{\mathbf{n}}
\newcommand{\C}{\mathbb{C}}
\newcommand{\Q}{\mathbb{Q}}
\renewcommand{\phi}{\varphi}
\newcommand{\limn}{\lim_{n\to \infty}}
\newenvironment{bp}{\color{blue}\begin{proof}}{\end{proof}}
\newenvironment{bs}{\color{blue}\begin{proof}[Solution]}{\end{proof}}
\renewcommand{\emph}[1]{\textbf{#1}}
\renewcommand{\bar}[1]{\overline{#1}}
\renewcommand{\Im}{\text{Im}}
\renewcommand{\Re}{\text{Re}}
\newcommand{\dudn}{\frac{\partial u}{\partial n}}

\newtheorem*{thm*}{Theorem}

\begin{document}

{ \large \textbf{\textsc{Math 5587 -- Homework 8 (Due Thursday Nov 10)}}}
\thispagestyle{empty}
\vspace{0.1in}

In each problem, $D \subset \R^n$ is open and bounded, and $n=3$ unless otherwise specified.
\begin{enumerate}
\item Let $u\in C^2(\bar{D})$ be a solution of
\begin{equation}\label{eq:PoissonD2}
\left.
\begin{aligned}
u-\Delta u &= f&&\text{in } D\\
u &=0&&\text{on } \partial D.
\end{aligned}\right\}
\end{equation}
Use energy methods to show that
\begin{equation}\label{eq:energy}
\iiint_D u^2 + \|\nabla u\|^2 \, d\x \leq \iiint_D f^2 \, d\x.
\end{equation}
Indicate how this is a stability estimate for \eqref{eq:PoissonD2}. [Hint: Multiple the PDE by $u$, integrate both sides over $D$, and then use Green's first identity. Use Cauchy's inequality $2ab \leq a^2 + b^2$ on the right hand side.]
\item Use energy methods to prove uniqueness of solutions $u\in C^2(\bar{D})$ of the Robin problem
\[\left. \begin{aligned}
-\Delta u(\x) &=f(\x)&& \text{if }\x\in D\\
\frac{\partial u}{\partial \n}(\x) + a(\x)u(\x)&=h(\x)&& \text{if } \x \in\partial D,\end{aligned}\right\}\]
provided $a>0$.
\item Consider the heat equation
\[\left\{ \begin{aligned}
u_t -\Delta u &=f&& \text{in } D\times (0,T]\\
u&=g&& \text{on } D\times\{t=0\}\\
u&=h&& \text{on } \partial D \times (0,T].\end{aligned}\right.\]
Prove uniqueness using energy methods. [Hint: Let $u$ and $v$ be two solutions and set $w=u-v$. Define the energy
\[e(t) = \iiint_D w(\x,t)^2 \, d\x.\]
Show that $e'(t) \leq 0$ and $e(0)=0$. Conclude that $e(t)=0$ for all $t$. You will need to use Green's identities.]
\item Let $u,v \in C^2(\bar{D})$ satisfy
\[-\Delta u + F(u) \leq -\Delta v + F(v) \ \ \text{ in } D,\]
where $F:\R \to \R$ is increasing, that is, $F(s)\leq F(t)$ whenever $s\leq t$. Prove the comparison principle: 
\begin{center}
If $u\leq v$ on $\partial D$ then $u \leq v$ everywhere in $D$. 
\end{center}
[Hint: Assume to the contrary that $u(\x) > v(\x)$ for some $\x\in D$, and define the \emph{open} set $D' = \{\x \in D \ | \ u(\x) > v(\x)\}$. Set $w:=u-v$ and show that $\Delta w \geq 0$ on $D'$ and $w \leq 0$ on $\partial D'$. Use the weak maximum principle on $D'$ to get a contradiction.]
\item Give an example in $n=1$ dimension to show that the comparison principle from Problem 4 may not hold if $F$ is not increasing. [Hint: Consider the eigenvalue problem $u''(x) + u(x) = 0$ with $u(0)=u(\pi)=0$ that we encountered in separation of variables.]
\item Use the comparison principle from Problem 4 to prove uniqueness of solutions $u\in C^2(\bar{D})$ of
\begin{equation}\label{eq:Poisson_nonlinear}
\left.
\begin{aligned}
-\Delta u + F(u) &= f&&\text{in } D\\
u &=g&&\text{on } \partial D,
\end{aligned}\right\}
\end{equation}
where $F:\R \to \R$ is increasing.
\item A function $u \in C^2(\bar{D})$ is \emph{subharmonic} in $D$ if $-\Delta u \leq 0$ in $D$. Let $u(x,y)$ be subharmonic in an open set $D \subset \R^2$. 
\begin{enumerate}
\item Show that
\[u(x,y) \leq \frac{1}{2\pi} \int_0^{2\pi} u(x+a\cos\theta,y + a\sin\theta) \, d\theta,\]
for all $a>0$ such that the ball of radius $a$ centered at $(x,y)$ belongs to $D$. 
[Hint: It is enough to prove the result for $x=y=0$, by translation invariance. Use Poisson's integral formula to construct a harmonic function $v$ on the disk $x^2 + y^2 \leq a^2$ with boundary values $v = u$ for $x^2+y^2=a^2$. Then use the comparison principle from Problem 4 to show that $u \leq v$ in the disk.]
\item Integrate the expression above in polar coordinates to show that
\[u(x,y) \leq \frac{1}{\pi a^2} \int_0^{2\pi}\int_0^a u(x+r\cos\theta,y+r\sin\theta) \, rdr d\theta,\]
whenever the ball $B(\x,a)$ is contained in $D$. 
\end{enumerate}
\item Let $\{\alpha_n\}_{n=1}^\infty$ denote the positive solutions of the equation $\alpha\tan(\alpha) = 1$. Find the solution $u(x,y)$ of the boundary-value problem
\[\left\{\begin{aligned}
\Delta u &=0,&&&& 0 < x < 1, \ \ \ y>0\\
u_x(0,y) &=0,&&&& y>0\\
u(1,y) + u_x(1,y) &=0,&&&& y>0\\
u(x,0) &=1,&&&& 0 < x < 1\\
\end{aligned}\right.\]
that is bounded as $y \to \infty$.  [Hint: Use separation of variables and look for a series solution. Use the problems from HW7 to show that the eigenvalues are nonnegative and the eigenfunctions are orthogonal. You do not need to prove that the eigenfunctions are a complete orthonormal system, but do give formulas for the coefficients of the series.]
\end{enumerate}


\end{document}

