\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{\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{\subset}{\subseteq}
\renewcommand{\bar}[1]{\overline{#1}}
\renewcommand{\Im}{\text{Im}}
\renewcommand{\div}{\text{div}}
\renewcommand{\Re}{\text{Re}}
\newcommand{\dudn}{\frac{\partial u}{\partial n}}

\newtheorem*{thm*}{Theorem}

\begin{document}

{ \large \textbf{\textsc{Math 8385 -- Homework 1 (Due Oct 11)}}}
\thispagestyle{empty}
\vspace{0.1in}

\begin{enumerate}
\item Find and solve the Euler-Lagrange equation for the functional
\[I(u) = \int_0^{\log(2)} u(x)^2 + u'(x)^2 \, dx\]
subject to boundary conditions $u(0)=0$ and $u(\log(2))=1$, where $\log$ is the natural logarithm. 
\item Find the Euler-Lagrange equation for the functional
\[I(u) = \int_0^\pi u(x)^2 - (u'(x))^2 \, dx\]
subject to boundary conditions $u(0)=u(\pi)=0$. Then find \emph{all} solutions of the Euler-Lagrange equation (as a one-parameter family) and evaluate $I$ on all solutions.
%\item Let $|x| = \sqrt{x_1^2 + \cdots + x_n^2}.$
%\begin{enumerate}
%\item Show that for $x\neq 0$
%\[\frac{\partial }{\partial x_i} |x| = \frac{x_i}{|x|}.\]
%\item Show that for $x\neq 0$
%\[\frac{\partial^2 }{\partial x_i\partial x_j} |x| = \frac{\delta_{ij}}{|x|} - \frac{x_ix_j}{|x|^3},\]
%where $\delta_{ij}$ is the Kronecker delta defined by
%\[\delta_{ij} = \begin{cases}
%1,&\text{if } i=j\\
%0,&\text{if } i\neq j.
%\end{cases}\]
%\item Show that for $x\neq 0$
%\[\Delta |x| = \frac{n-1}{|x|}.\]
%\end{enumerate}
%\item Find all real numbers $\alpha$ for which $u(x) = |x|^\alpha$ is a solution of Laplace's equation
%\[\Delta u(x) = 0 \ \ \text{ for } x\neq 0.\]
\item Let $1 \leq p \leq \infty$. The $p$-Laplacian is defined by
\[\Delta_p u := \div\left(|\nabla u|^{p-2}\nabla u\right)\]
for $1 \leq p < \infty$, and 
\[\Delta_\infty u := \frac{1}{|\nabla u|^2}\sum_{i=1}^n\sum_{j=1}^n u_{x_ix_j}u_{x_i}u_{x_j}.\]
Notice that $\Delta_2 u = \Delta u$. A function $u$ is called $p$-\emph{harmonic} if $\Delta_p u=0$.  
\begin{enumerate}
\item Show that the Euler-Lagrange equation for the functional
\[I(u) = \int_U \frac{1}{p}|\nabla u(x)|^p -u(x)f(x)\, dx\]
is the $p$-Laplace equation 
\[-\Delta_p u = f \ \ \text{in } U.\]
\item Show that
\[\Delta_p u =|\nabla u|^{p-2} \left( \Delta u + (p-2)\Delta_\infty u\right). \]
\item Show that
\[\Delta_\infty u = \lim_{p\to\infty} \frac{1}{p}|\nabla u|^{2-p}\Delta_p u.\]
\end{enumerate}
%\item Let $1 \leq p \leq \infty$. Find all real numbers $\alpha$ for which the function $u(x) = |x|^\alpha$ is $p$-harmonic away from $x=0$.
\item Let $u,w \in C^2(\bar{U})$ where $U \subset \R^n$ is open and bounded. Show that for $1 \leq p < \infty$
\[\int_U u\,\Delta_p w \, dx = \int_{\partial U} u\,|\nabla w|^{p-2}\frac{\partial w}{\partial \nu} \, dS - \int_U |\nabla w|^{p-2}\nabla u\cdot \nabla w\, dx.\]
[Hint: Use one of the integration by parts formulas in the appendix of the course notes.]
\item Find the Euler-Lagrange equation for the functional
\[I(u) = \int_U (\Delta u(x))^2 \, dx.\]
[Hint:  Proceed as in the proof of the Euler-Lagrange equation from class. That is, let $\phi$ be smooth with compact support in $U$ and compute
\[\frac{d}{dt}\Big\vert_{t=0} I(u+t\phi)=0.\]
Use integration by parts and the vanishing lemma to find the Euler-Lagrange equation.]
\item Consider the constrained problem
\[\min_{\substack{u:U \to \R \\ u=0 \text{ on } \partial U} } \int_U |\nabla u|^2 \, dx \ \ \text{subject to} \ \ \int_U u^2 \, dx = 1.\]
Show that any minimizer is a solution of the eigenvalue problem
\begin{equation}\label{eq:eigenvalue}
\left\{\begin{aligned}
-\Delta u &= \lambda u&&\text{in }U\\ 
u &=0&&\text{on }\partial U,
\end{aligned}\right.
\end{equation}
where $\lambda > 0$ is given by
\[\lambda = \int_U |\nabla u|^2\, dx.\]
[Hint: This does not require a Lagrange multiplier. Let $u$ be a minimizer of the constrained problem, let $\varphi \in C^\infty_c(U)$, and consider the variation
\[t \mapsto w(x,t):= \frac{u(x) + t\varphi(x)}{\|u + t\varphi\|_{L^2(U)}}.\]
Since $\displaystyle \int_U w(x,t)^2 \, dx =1$, we have $\displaystyle \frac{d}{dt}\Big\vert_{t=0} \int_U|\nabla w(x,t)|^2\,dx=0$. Complete the argument from here.]
\item  Recall the minimal surface equation
\[\div\left( \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} \right) = 0 \ \ \ \text{in }U.\]
\begin{enumerate}
\item Show that the plane
\[u(x) = a\cdot x + b\]
solves the minimal surface equation on $U=\R^n$, where $a\in \R^n$ and $b \in \R$.
\item Show that for $n=2$ the Scherk surface
\[u(x) = \log\left(\frac{\cos(x_1)}{\cos(x_2)} \right)\]
solves the minimal surface equation on the box $U=(-\tfrac{\pi}{2},\tfrac{\pi}{2})^2$.
\end{enumerate}
\end{enumerate}

\end{document}

