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\centerline{\textbf{Note:} Please choose a paper for your term end project/presentation by Friday Oct 5. \\}
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{ \large \textbf{\textsc{Math 8590 -- Homework 2 (Due Friday Oct 5)}}}
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Please hand in your solution to 1 problem from those below.

Let $U\subset \R^n$ be open.
\begin{enumerate}
\item 
\begin{enumerate}
\item Let $u,v \in \usc(\bar{U})$. Suppose that $w:=u$ and $w:=v$ are viscosity solutions of 
\begin{equation}\label{eq:H}H(D^2w,Dw,w,x) \leq 0 \ \ \text{in U}.\end{equation}
Show that $w(x):=\max\{u(x),v(x)\}$ is a viscosity solution of \eqref{eq:H} (i.e., the pointwise maximum of two subsolutions is again a subsolution).
\item Let $u,v \in \lsc(\bar{U})$. Suppose that $w:=u$ and $w:=v$ are viscosity solutions of 
\begin{equation}\label{eq:H2}H(D^2w,Dw,w,x) \geq 0 \ \ \text{in U}.\end{equation}
Show that $w(x):=\min\{u(x),v(x)\}$ is a viscosity solution of \eqref{eq:H2}.
\end{enumerate}
\item For each $k\in \N$, let $u_k \in C(U)$ be a viscosity solution of
\[H(D^2u_k,Du_k,u_k,x) = 0 \ \ \text{in } U.\]
Suppose that $u_k \to u$ locally uniformly on $U$ (this means $u_k \to u$ uniformly on every $V\subset\subset U$). Show that $u$ is a viscosity solution of
\[H(D^2u,Du,u,x) = 0 \ \ \text{in } U.\]
Thus, viscosity solutions are stable under uniform convergence. (We will see shortly that viscosity solutions are stable under even weaker types of convergence.)
\item Suppose that $H=H(p,x)$ is continuous and $p\mapsto H(p,x)$ is \emph{convex} for any fixed $x$. Let $u \in C^{0,1}_{loc}(U)$ satisfy
\[\lambda u(x) + H(Du(x),x) \leq 0 \ \ \text{ for a.e. } x \in U,\]
where $\lambda \geq 0$. Show that $u$ is a viscosity solution of 
\[\lambda u + H(Du,x) \leq 0 \ \ \text{in } U.\]
Give an example to show that the same result does not hold for supersolutions. [Hint: Mollify $u$: $u_\eps:=\eta_\eps * u$. For $V \subset \subset U$, use Jensen's inequality to show that
\[\lambda u_\eps(x) + H(Du_\eps(x),x) \leq h_\eps(x) \ \ \text{for all } x \in V\]
and $\eps>0$ sufficiently small, where $h_\eps \to 0$ uniformly on $V$. Then apply an argument similar to problem 2.]
\item Let $1 < p < \infty$ and define
\[|x|_p :=\left(\sum_{i=1}^n |x_i|^p\right)^\frac{1}{p}.\]
Assume $U\subset \R^n$ is open, bounded, and path connected with Lipschitz boundary $\partial U$, and let $f:\bar{U} \to \R$ be continuous and positive. Show that there exists a unique viscosity solution $u \in C(\bar{U})$ of the p-eikonal equation
\[\text{(P)} \ \ \left\{\begin{aligned}
|Du|_p &=f&&\text{in } U\\
u&=0&&\text{on } \partial U.
\end{aligned}\right.\]
[Hint: Construct $u$ as the value function
\[u(x) = \inf\{T(x,y) \, : \, y \in \partial U\},\]
where 
\[T(x,y) = \inf\left\{\int_0^1 f(\w(t))|\w'(t)|_q\,dt\, : \, \w \in C^1([0,1];\bar{U}), \, \w(0)=x, \, \w(1)=y\right\},\]
and $q$ is the H\"older conjugate of $p$, i.e., $\tfrac{1}{p} + \tfrac{1}{q} = 1$. Don't worry about exactly computing the form of $H$. Instead show that any solution of $H=0$ is also a solution of (P).]
\end{enumerate}
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