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\begin{document}

{ \large \textbf{\textsc{Math 8385 -- Homework 2A (Due Friday November 22)}}}
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\vspace{0.1in}

\noindent Let $u\in H^1(U)$ be a weak solution of
\[\sum_{i,j=1}^n\frac{\partial}{\partial x_j}(a^{ij}u_{x_i}) = 0 \ \ \ \text{in } U.\]
That is, for every $v\in H^1_0(U)$ we have
\[\int_U \sum_{i,j=1}^na^{ij}u_{x_i}v_{x_j}\, dx = 0.\]
Assume the $a^{ij}:U\to \R$ are bounded and measurable, and satisfy the ellipticity condition
\[\theta|\eta|^2 \leq \sum_{i,j=1}^n a^{ij}(x)\eta_i \eta_j \leq \Theta|\eta|^2 \ \ \ \ \ \ (\forall x\in U, \eta \in \R^n),\]
where $0 < \theta \leq \Theta$.
In this homework, you will show that for $n=2$ we have $u\in C^{0,\gamma}_{loc}(U)$ for some $\gamma>0$. This is the interior version of the de Giorgi-Nash-Moser theory.
\begin{enumerate}
\item Let $x_0 \in U$ and $r>0$ such that $B(x_0,2r) \subset U$. 
\begin{enumerate}
\item Show that there exists a constant $C>0$, depending only on $\theta$ and $\Theta$, such that
\begin{equation}
\int_{B(x_0,r)} |Du|^2 \, dx \leq \frac{C}{r^2}\int_{B(x_0,2r)\setminus B(x_0,r)} |u-a|^2 \, dx,
\end{equation}
where $a$ is any real number. [Hint: Let $\zeta \in C^\infty(\R^n)$ be a smooth cutoff function satisfying $\zeta\equiv 1$ on $B(x_0,r)$, $\zeta\equiv 0$ on $\R^n\setminus B(x_0,2r)$, $0 \leq \zeta \leq 1$, and $|D\zeta| \leq \tfrac{2}{r}$. Substitute $v=(u-a)\zeta^2$ into the definition of weak solution.]
\item Verify the Poincar\'e inequality 
\[\int_{B(x_0,2r)\setminus B(x_0,r)} |u-a|^2 \, dx \leq C r^2 \int_{B(x_0,2r)\setminus B(x_0,r)}|Du|^2 \, dx\]
holds for
\[a = \dashint_{B(x_0,2r)\setminus B(x_0,r)} u \, dx.\]
\item Combine parts (a) and (b) to deduce
\[\int_{B(x_0,r)} |Du|^2 \, dx \leq \frac{C}{C+1} \int_{B(x_0,2r)} |Du|^2 \, dx,\]
where $C>0$ depends only on $\theta$ and $\Theta$.
[Hint: After applying Poincar\'e's inequality, add $C\int_{B(x_0,r)}|Du|^2 \, dx$ to both sides the equation. This is known as the ``hole-filling'' trick.]
\end{enumerate}
\item Define
\[\phi(r) := \int_{B(x_0,r)} |Du|^2 \, dx.\]
By Part 1, there exists $0 < \eta < 1$, depending only on $\theta$ and $\Theta$, such that
\begin{equation}\label{eq:key}
\phi\left( \frac{r}{2} \right) \leq \eta\phi(r)  \ \ \ \ \text{ for all } 0 <r < r_0,
\end{equation}
where $r_0 = \dist(x_0,\partial U)$. 
\begin{enumerate}
\item Show that there exists $0 < \lambda \leq 1$, depending only on $\eta$, such that
\begin{equation}\label{eq:conc}
\phi(r) \leq  \frac{\varphi(r_0)}{\eta}\left(\frac{r}{r_0}\right)^\lambda  \ \ \ \ \text{ for all } 0 <r < r_0.
\end{equation}
\item Use (a) and Poincar\'e's inequality for a ball to show that
\begin{equation}\label{eq:p}
\dashint_{B(x_0,r)} |u - (u)_{x_0,r}|^2  \, dx \leq Cr^{\lambda + 2-n}
\end{equation}
for all $0 < r < r_0$,  where $C$ depends on $\theta$, $\Theta$, and $r_0=\dist(x_0,\partial U)$. Recall $(u)_{x_0,r} = \dashint_{B(x_0,r)} u \, dx$.
\end{enumerate}
\item Assume $n=2$. Use Part 2 to prove that $u \in C^{0,\gamma}_{loc}(U)$ for $\gamma=\lambda/2$. This establishes the local H\"older continuity portion of the de Giorgi-Nash-Moser theory in dimension $n=2$. \textbf{Hint:} Follow the steps below.
\begin{enumerate}
\item Fix $\eps>0$ and define
\[U_\eps = \{x \in U \, : \, \dist(x,\partial U) > \eps\}.\]
Show that for any $x_0 \in U_\eps$ and $0 < s < t < \eps$
\[s^2|(u)_{x_0,s} - (u)_{x_0,t}|^2 \leq C(s^{\lambda+2} + t^{\lambda+2}).\]
\item Let $x_0 \in U_\eps$, $r<\eps$, and define $r_j = r2^{-j}$ and $a_j = (u)_{x_0,r_j}$. Use part (a) to show that
\[|u(x_0) - (u)_{x_0,r}| \leq \sum_{j=0}^\infty |a_{j+1}-a_{j}| \leq Cr^\gamma,\]
for almost every $x_0 \in U_\eps$, where $\gamma=\lambda/2$.
\item Conclude from part (b) that $u \in C(\bar{U_\eps})$ (provided we identify $u$ with its continuous version). [Hint: $(u)_{x,r}$ is a continuous function of $x$ for every $r>0$.]
\item Show that $u \in C^{0,\gamma}(\bar{U_\eps})$. [Hint: Let $x,y \in U_\eps$ with $r:=|x-y| < \eps$. Write
\[|u(x)-u(y)| \leq |u(x)-(u)_{x,r}| + |(u)_{x,r} - (u)_{y,r}| + |u(y) - (u)_{y,r}|.\]
Estimate the 1st and 3rd terms with part (b). For the second term, mimic the argument used at the end of the proof of Morrey's inequality (Theorem 4 in Evans Section 5.6.2).]
\end{enumerate}
\end{enumerate}

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