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{ \large \textbf{\textsc{Math 8590 -- Homework 1 (Due Friday Sept 21)}}}
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Please hand in your solution to 1 problem from those below.
\begin{enumerate}
\item Let $u \in \text{USC}(\R^n)$ and define
\[A = \Big\{x \in \R^n \, : \, \exists \phi\in C^\infty(\R^n),\, u-\phi \text{ has a local max at } x\Big\}.\]
Show that $A$ is dense in $\R^n$. [Hint: Let $x_0 \in \R^n$ and $\eps>0$, and consider the maximum of $u-\phi$ over $B(x_0,1)$, where $\phi(x) := \frac{|x-x_0|^2}{\eps}$. Send $\eps\to 0$. You should recall that upper semicontinuous functions assume their maximums over compact sets (why?).]
\item Show that $u(x)=x$ is a viscosity solution of $u'=1$ on the interval $(0,1]$, but is \emph{not} a viscosity solution of $u'=1$ on the interval $[0,1)$. [Hint: Examine the subsolution condition at $x=0$. This exercise shows that smooth solutions need not be viscosity solutions at boundary points.]
\item Let $u:(0,1)\to \R$ be continuous. 
\begin{enumerate}
\item Show that $u$ is nondecreasing on $(0,1)$ if and only if $u$ is a viscosity solution of $u'\geq 0$ on $(0,1)$. [Hint: For the hard direction, suppose that $u'\geq 0$ in the viscosity sense on $(0,1)$, but $u$ is not nondecreasing on $(0,1)$. Show that there exists $0<x_1<x_2<x_3<1$ such that $u(x_3)<u(x_2)<u(x_1)$. Construct a test function $\phi \in C^\infty(\R)$ with $\phi'<0$ such that $\phi$ touches $u$ from below somewhere in the interval $(x_1,x_3)$. Drawing a picture might help.]
\item Show that $u$ is convex on $(0,1)$ if and only if $u$ is a viscosity solution of $-u''\leq 0$ on $(0,1)$. Show that in general, convex functions are not viscosity solutions of $u''\geq 0$. [Hint: The hint for the hard direction is similar to part (a). Suppose that $-u''\leq 0$ on $(0,1)$ but $u$ is not convex on $(0,1)$. Then there exists $0 < x_1 < x_2<1$ and $\lambda \in (0,1)$ such that
\[u(\lambda x_1 + (1-\lambda)x_2) > \lambda u(x_1) + (1-\lambda)u(x_2) .\] 
Construct a test function $\phi \in C^\infty(\R)$ with $\phi''<0$ such that $\phi$ touches $u$ from above somewhere in the interval $(x_1,x_2)$.]
\end{enumerate}
\item Let $U \subset \R^n$ be open. Suppose that $u\in C(U)$ satisfies
\[u(x) = \dashint_{B(x,\eps)} u \, dy + o(\eps^2) \ \ \text{as } \eps \to 0^+\]
for every $x \in U$. Note this is an asymptotic version of the mean value property. Show that $u$ is a viscosity solution of 
\[-\Delta u = 0 \ \ \text{in } U.\]
[Hint: Show that for every $\phi\in C^\infty(\R^n)$ 
\[-\Delta \phi(x) = 2(n+2)\dashint_{B(x,\eps)} \frac{\phi(x) - \phi(y)}{\eps^2} \, dy  + o(1) \ \ \text{ as } \eps \to 0^+.\]
To do this, write a second order Taylor expansion for $\phi$ at $x$
\[\phi(y) = \phi(x) + D\phi(x)\cdot (y-x) + \frac{1}{2}(y-x)^TD^2\phi(x)(y-x) + o(|y-x|^2) \ \ \text{ as } y \to x,\]
and average both sides over the ball $B(x,\eps)$. Then verify the viscosity sub- and supersolution properties directly from the definitions.]
\end{enumerate}
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