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\begin{document}
\begin{center}
 PDE Qual Prep Seminar\\
 Xavier Ramos Oliv\'e\\
 Summer 2018, UC Riverside
\end{center}


\section*{207A material}

\subsection*{ODEs}

\begin{Prob}
 Suppose $f(t,y): \R\times \R^n \rightarrow \R$ is continuous on $\mathcal{R}$: $t_0\leq t\leq t_0+a$, $|y-y_0|\leq b$, and uniformly Lipschitz continuous with respect to $u$ in $\mathcal{R}$. Let $M$ be an upper bound for $|f(t,y)|$ on $\mathcal{R}$ and $\alpha := \min (a,b/M)$. Prove that
 
 $$\frac{dy}{dt} = f(t,y)\ \ \ y(t_0) = y_0$$

 has a solution on $[t_0, t_0+\alpha]$.
 \vskip 1em
 
 [Source: HW1 2015, 2017, Midterms 2015, 2017, A. Moradifam]
\end{Prob}


\subsection*{Laplace's equation}

\begin{Prob}
 Prove that if $u\in \mathcal{C}^2(\Omega)$ is harmonic, then
 
 $$u(x) = \fint_{\partial B(x,r)} u dS,$$
 
 for all $B(x,r)\subset \Omega$.
 \vskip 1em
 [Source: Midterm 2015, A. Moradifam]
\end{Prob}

\begin{Prob}
(Harnack's Inequality) Let $V\subset \subset \Omega$. Prove that there exists a constant $C$, depending only on $V$, such that

$$\sup_V u \leq C\inf_V u$$

for all nonnegative harmonic functions $u$ in $\Omega$. 

\vskip 1em
 [Source: Midterm 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
 Let $u\in \mathcal{C}^2(\Omega)\cap \mathcal{C}(\bar{\Omega})$ solve the Laplace equation
 $$-\Delta u = f\text{ on }\Omega,$$
 with $u=g$ on $\partial \Omega$. Prove that there exists a constant $C$, independent of $u$, such that 
 $$\max_{\Omega} |u| \leq C (\max_{\partial \Omega} |g| + \max_{\Omega} |f|).$$
 \vskip 1em
 [Source: HW3 2015, 2017 (Evans Ch. 2), Midterms 2015, 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
Let $u_1, u_2 \in \mathcal{C}^2(\Omega)$ solve the Laplace equation

$$\begin{cases}
   \Delta u = f & \text{in }\Omega\\
   \frac{\partial u}{\partial \nu} = g & \text{on }\partial \Omega.
  \end{cases}
$$
Prove that $u_2 - u_1 = c$, where $c$ is a constant.

\vskip 1em
 [Source: Final 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
Let
$$\mathcal{A}:= \{ \omega \in \mathcal{C}^2 (\bar{\Omega})\colon \omega = g\text{ on } \partial \Omega\}$$
and define $I: \mathcal{A}\rightarrow \R$ by
$$I(\omega):= \int_\Omega \frac{|D\omega|^2}{2} - \omega fdx.$$
Prove that $u\in \mathcal{A}$ is a minimizer of $I$ in $\mathcal{A}$ if and only if $u$ solves the Laplace equation
$$-\Delta u = f$$

\vskip 1em
 [Source: Final 2015, Midterm 2017, A. Moradifam]
\end{Prob}


\begin{Prob}
Suppose $u\in \mathcal{C}^2(\bar{\Omega})$ solves the equation

$$\begin{cases}
   \Delta u = g, & \text{in }\Omega\\
   u= \frac{\partial u}{\partial \nu} = 0, &\text{on }\partial \Omega , 
  \end{cases}
$$

where $\nu$ is the outward unit normal vector on $\partial \Omega$. Prove that

$$\int_\Omega g\varphi dx = 0$$

for every harmonic function $\varphi \in \mathcal{C}(\bar{\Omega})$. 

\vskip 1em
 [Source: Midterm 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
 Let $(u_n)_{n=1}^\infty$ be a sequence of harmonic functions defined on an open bounded subset $U$ of $\R^d$, $d\geq 2$. Assume that $u_n\rightarrow u$ uniformly on $U$. Prove that $u$ is harmonic on $U$.
 
 \vskip 1em
 [Source: Qual 2015, Final 2014, J. Kelliher]
\end{Prob}


\subsection*{Heat equation}


\begin{Prob}
Let $u\in \mathcal{C}_1^2(\Omega_T)$ solve the heat equation. Prove that
$$u(x,t) = \frac{1}{4r^n}\int \int_{E(x,t;r)} u(y,s)e^{-\frac{|x-y|^2}{(t-s)^2}} dyds,$$
for every heat ball $E(x,t;r)\in \Omega_T$. Note that

$$E(x,t;r) = \{(y,s)\in \R^{n+1}\colon s\leq t, \frac{1}{(4\pi(t-s))^{n/2}}e^{-\frac{|x-y|^2}{4(t-s)}} \geq \frac{1}{r^n}\}$$

and 

$$\frac{1}{r^n}\int \int_{E(r)} \frac{|y|^2}{s^2}dyds = 4$$

where $E(r) = E(0,0;r)$.

\vskip 1em
 [Source: Final 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
 (Strong maximum principle for the heat equation) Let $\Omega \subset \R^n$ and $T>0$ and suppose $u\in \mathcal{C_1^2}(\Omega_T)\cap \mathcal{C}(\bar{\Omega_T})$ solves the heat equation in $\Omega_T$. Prove that if $\Omega$ is connected and there exists $(x_0, t_0)\in \Omega_T$ such that
 
 $$u(x_0, t_0) = \max_{\bar{\Omega_T}} u,$$
 
 then $u$ is constant in $\bar{\Omega_{t_0}}$. {\bf Hint:} Use the mean-value property for the heat equation:
 
 $$u(x,t) = \frac{1}{4r^n}\int \int_{E(x,t;r)} u(y,s) \frac{|x-y|^2}{(t-s)^2} dyds,$$
 
 where $E(x,t;r)$ is the heat ball centered at $(x,t)$.
 
 \vskip 1em
 [Source: Final 2015, A. Moradifam]
\end{Prob}

\begin{Prob}
 Let $g\in \mathcal{C}(\R^n)\cap L^\infty (\R^n)$ and define
 
 $$u(x,t) = \int_{\R^n} \Phi (x-y,t)g(y)dy$$
 
 where $\Phi$ is the fundamental solution of the heat equation ($\Phi(x,t) = \frac{1}{(4\pi t)^{n/2}}e^{\frac-{|x|^2}{4t}}$). Prove that $u$ solves the heat equation and 
 $$\lim_{(x,t)\rightarrow (x_0, 0^+)} u(x,t) = g(x_0),$$
 for all $x_0 \in \R^n$.
 \vskip 1em
 [Source: Midterm 2015, Final 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
(Backwards Uniqueness for the heat equation) Suppose $u,\tilde{u}\in \mathcal{C}^2 (\bar{U}_T)$ solve the heat equation

$$\begin{cases}
   u_t -\Delta u = 0 & \text{in }\Omega \times (0,T]\\
   u=g & \text{on } \partial \Omega \times [0, T],
  \end{cases}
$$

Prove that if $u(x,T) = \tilde{u}(x,T)$ ($x\in \Omega$), then 

$$u=\tilde{u} \text{ in }\Omega \times (0,T].$$

\vskip 1em
 [Source: Qual 2016, A. Moradifam]
\end{Prob}

\begin{Prob}
 Let $u(x,t) = v\left( \frac{x^2}{t}\right)$. (a) Show that 
 
 $$u_t - u_{xx} = 0$$
 
 if and only if
 
 $$4z\frac{\partial^2v}{\partial z^2} + (2+z)\frac{\partial v}{\partial z}(z) = 0,\ z>0.$$
 (b) Use part (a) to obtain the fundamental solution of the heat equation in dimension $n=1$.
 \vskip 1em
 [Source: HW4 2015, 2017 (modified from Evans Ch. 2), Final 2015, A. Moradifam]
\end{Prob}

\subsection*{Wave equation}

\begin{Prob}
(a) Show that the general solution of the PDE $u_{xy}= 0$ is 
$$u(x,y) = F(x) + G(y)$$
for arbitrary functions $F$ and $G$. \\
(b) Using the change of variable $\xi = x+t$, $\eta = x-t$, show that $u_{\xi\eta} = 0$ if and only if 
$$u_{tt}-u_{xx} = 0.$$
(c) Use (a) and (b) to prove that the general solution of the wave equation in dimension one is given by
$$u(x,t) = \frac{1}{2}[g(x+t)+g(x-t)] + \frac{1}{2}\int_{x-t}^{x+t} h(y) dy$$

\vskip 1em
 [Source: Final 2015, Final 2017 (Evans Ch. 2), A. Moradifam]
\end{Prob}

\begin{Prob}
(Finite speed of propagation for solutions of the wave equation) Suppose $u\in \mathcal{C}^2 (\R^n \times (0, \infty))$ solves the wave equation. Prove that if $u = u_t = 0$ on $B(x_0, t_0)\times \{ 0 \}$, then $u \equiv 0$ within the cone

$$C = \{(x,t)\colon 0\leq t\leq t_0,\ |x-x_0|\leq t_0-t\}$$

{\bf Hint:} consider the energy functional 

$$e(t) = \frac{1}{2}\int_{B(x_0, t_0-t)} u_t^2(x,t)+ |Du(x,t)|^2 dx\ \ \ (0\leq t\leq t_0)$$ 
\vskip 1em
 [Source: Final 2015, Qual 2016, Final 2017, A. Moradifam]
\end{Prob}

\begin{Prob}
 Assume that $u$ is a smooth solution to the wave equation in $\R^d$, $d\geq 1$, in the form
 $$\partial_{tt} u = \Delta u.$$
 Define the energy-density $e(t,x):= \frac{1}{2}(\partial_t u)^2 + \frac{1}{2} |\nabla u|^2$. 
 
 \begin{enumerate}
  \item [(a)] Show that for $d=1$, $e$ itself satisfies the one-dimensional wave equation.
  \item [(b)] Show that $e$ does not, in general, satisfy the $d$-dimensional wave equation for $d\geq 2$. 
  \item [(c)] Suppose that $u\in \mathcal{C}^\infty (\R\times \R^d)$, $d\geq 1$, and compactly supported in space for all $t$. Show that
  
  $$\int_{\R^d} (\partial_{tt}e(t,x)-\Delta e(t,x))dx = 0$$
 \end{enumerate}

 \vskip 1em
 [Source: Qual 2015, J. Kelliher]
\end{Prob}



\subsection*{Other equations}

\begin{Prob}
Write down an explicit formula for a function $u$ solving the equation

$$\begin{cases}
   u_t-b\cdot \nabla u +cu = 0 & \text{in } \R^n \times (0,\infty)\\
   u=g & \text{on } \R^n \times \{ t=0 \},
  \end{cases}
$$

where $c\in \R$ and $b\in \R^n$ are constant.
\vskip 1em
 [Source: HW2 2015, 2017 (Evans Ch.2), Qual 2016, A. Moradifam]
\end{Prob}

\begin{Prob}
Suppose $u$ is an integral solution of $$u_t +F(u)_x = 0,$$ and $u\in \mathcal{C}^1 ((\R\times (0, \infty))\setminus \Gamma)$, where $\Gamma$ is a one dimensional curve. Prove that if $u$ is discontinuous on $\Gamma$, then it should satisfy the jump condition 

$$(F(u_l)-F(u_r))\eta^1 + (u_l-u_r)\eta^2 = 0,$$

where $(\eta^1, \eta^2)(x)$ is the normal to the curve $\Gamma$ at $x$. 

\vskip 1em
 [Source: Final 2015 (Evans 3.4.1), A. Moradifam]
\end{Prob}

\newpage
\section*{207B material}

\subsection*{Flow maps}

\begin{Prob}
 Let ${\bf u}(t,x)$ be a time-varying vector field on $\R^d$. 
 \begin{enumerate}
  \item[(a)] Define what it means for $X:\R\times \R^d \rightarrow \R^d$ to be a flow map for $\bf u$.
  \item[(b)] Assume that ${\bf u}_1$, ${\bf u}_2$ are Lipschitz continuous vector fields with Lipschitz constants $M_1$, $M_2$ for all $t\in \R$; that is, $|{\bf u}_j (t,x) - {\bf u}_j(t,y)|\leq M_j |x-y|$, $j=1,2$, for all $t\in \R$, $x,y\in \R^d$. Let $X$ be the flow map for ${\bf u}:= {\bf u}_1 + {\bf u}_2$ and $X_j$ be the flow map for ${\bf u}_j$, $j=1,2$ (you do not need to prove that such flow maps exist and are unique). Show that 
  $$|X(t,x)- X_1(t,x)|\leq M_2 te^{M_1t}.$$
 \end{enumerate}
 
 \vskip 1em
 [Source: Qual 2016, J. Kelliher]
\end{Prob}

\begin{Prob}\label{problem1}
 Let ${\bf v}(t,x)$ be a time-varying vector field on $\R^d$. 
 \begin{enumerate}
  \item [(a)] Define what it means for $X:\R\times \R^d \rightarrow \R^d$ to be a flow map for $\bf v$. 
  \item [(b)] Suppose that $\bf v$ is uniformly Lipschitz-continuous in space; that is, 
  $$|{\bf v}(t,x) - {\bf v}(t,y)| \leq M |x-y| \text{ for all }x,y\in \R^d$$
  for some $M>0$. Prove that there exists a flow map for $\bf v$. 
  \item [(c)] Prove that the flow map for $\bf v$ is unique.
 \end{enumerate}
 
 \vskip 1em
 [Source: Qual 2015, J. Kelliher]
\end{Prob}

\begin{Prob}
Let $A(t)$ be a continuous function from $t$ in $\R$ to the space of square, real-valued matrices.
\begin{enumerate}
 \item [(a)] Show that for every solution of the (non-autonomous) linear system, ${\bf \dot{x}} = A(t) {\bf x}$, we have
 $$\|{\bf x}(t)\|\leq \|{\bf x}(0)\| e^{\int_0^t \|A(s)\|ds},$$
 where $\|A(s)\|$ is the operator norm and $\|{\bf x}(t)\|$ is the usual Euclidean norm.
 \item [(b)] Show that if $\int_0^t \|A(s)\|ds <\infty$ then every solution, ${\bf x}(t)$, has a finite limit as $t\rightarrow \infty$.
\end{enumerate}

 \vskip 1em
 [Source: Final 2014, J. Kelliher]
\end{Prob}



\subsection*{Method of characteristics}

\begin{Prob}
In this problem, we will seek a solution in the upper half plane to 

$$\begin{cases}
   -y\partial_x u + x\partial_y u = au & \text{for }x\in \R, y\geq 0,\\
   u(x,0) = \psi (x) & \text{for } x\geq 0,
  \end{cases}
$$

where $a$ is a real constant and $\psi: [0,\infty) \rightarrow \R$ is differentiable with $\psi (0) = \psi^\prime(0) = 0$. 

\begin{enumerate}
 \item[(a)] What are the characteristic equations for this PDE?
 \item[(b)] Solve the characteristic equations.
 \item[(c)] Use the solution in (b) to the characteristic equations to obtain an explicit solution to the PDE in the form $u=u(x,y)$.
 \item[(d)] Show that any compactly supported, differentiable solution to our PDE is unique whenever $a\leq 0$.
 {\bf Hint:} rewrite the PDE in the form ${\bf x}^\perp \cdot \nabla u = au$, where ${\bf x}^\perp = (-y, x)$, and make an energy argument. (The fact that $\rm{ div}\ {\bf x}^\perp = 0$ will be very useful).
\end{enumerate}

 \vskip 1em
 [Source: Midterm 2018, J. Kelliher]
\end{Prob}

\begin{Prob}
 Consider Burger's equation on the half-line, $r\in (0,\infty)$, with a non-linear forcing term: 
 
 $$\begin{cases}
    \partial_t v + v\partial_r v = -\frac{v^2}{r} & \text{for } t\in (0,\infty),\ r\in (0,\infty),\\
    v=v_0 & \text{for } t=0,\ r\in (0,\infty).
   \end{cases}
$$
 Assume that $v_0\in \mathcal{C}^\infty [0,\infty)$ with $v_0(0) = 0$ [note this implies that $\lim_{r\rightarrow 0^+} v^2(r) /r = 0$]. 
 
 In this problem, you will be asked to use the method of characteristics. As in any such problem, it is always possible that the characteristic projections will collide at some time, $T_1 \geq 0$, meaning that if $T\geq 0$ is the time of existence of a classical solution, then $T\leq T_1$.
 
 But because we are working on a half-line, a classical solution can also cease to exist because a characteristic projection reaches $r=0$ for some $t=T_2>0$, taking it outside the domain of solution. This means that also $T\leq T_2$, and hence $T = \min \{T_1, T_2\}$.
 
 \begin{enumerate}
  \item[(a)] Show that if we parametrize each characteristic by time and write each characteristic in the form $t\mapsto (r(t), v(r(t)))$ with $r(0) = r_0$, then:
  
  \begin{align*}
r(t)^2 &= r_0^2 +2r_0v_0(r_0)t,\\  
v(t,r(t)) &= \frac{r_0v_0(r_0)}{r(t)}
  \end{align*}
  \item[(b)] Show that 
  
  $$T_1 = \inf_{r_0>s_0>0} \frac{r_0^2-s_0^2}{2(s_0v_0(s_0)-r_0v_0(r_0)}$$
  
  \item[(c)] Show that 
  
  $$T_2 = \inf_{r_0>0} \left( -\frac{r_0}{2v_0(r_0)}\right)$$

 \end{enumerate}

 \vskip 1em
 [Source: Midterm 2018, J. Kelliher]
\end{Prob}

\begin{Prob}
 \begin{enumerate}
  \item [(a)] In the plane, solve the equation $y \frac{\partial u}{\partial x} - 4x\frac{\partial u}{\partial y} = 0$ with the condition that $u(0,y) = f(y)$ for all $y\in \R$. Assume that $f\in \mathcal{C}^\infty (\R)$ and find only $\mathcal{C}^\infty (\R^2)$ solutions.
  \item [(b)] Is the problem solvable uniquely in the full plane for all $f\in \mathcal{C}^\infty (\R^2)$? If not, what additional property or properties of $f$ is or are required for the problem to be solvable in the full plane? (Remember that we seek only $\mathcal{C}^\infty$ solutions).
 \end{enumerate}

 
 
 \vskip 1em
 [Source: Qual 2016, J. Kelliher]
\end{Prob}

\begin{Prob}
 Let ${\bf v}(t,x)$ be as in problem \ref{problem1} above in part (b), and suppose that $\rho = \rho (t,x)$ solves
 
 \begin{align*}
  \partial_t \rho + {\bf v}\cdot \nabla \rho = \rho^2,\\
  \rho (0) = \rho_0,
 \end{align*}

 where $\rho_0$ is continuous and bounded on $\R^d$. Suppose that $\rho_0 (x) \leq M$ for all $x\in \R^d$. Express the maximum possible time of existence of $\rho$ in terms of M. (You need not actually prove existence however.)
 
 {\bf Hint: }Use the result of the problem \ref{problem1} above - you can do so even if you didn't solve it.
 \vskip 1em
 [Source: Qual 2015, J. Kelliher]
\end{Prob}


\subsection*{Fourier Transform}

\begin{Prob}
 Solve the following initial value problem using the Fourier transform:
 $$\begin{cases}
    \partial_tt u +c^2\partial_{xxxx}u = 0 & \text{in }(0,\infty) \times \R,\\
    u=f, \partial_t u = g & \text{on } \{0\}\times \R.
   \end{cases}
$$
 Give your answer in the form of an inverse Fourier transform.
 \vskip 1em
 [Source: Qual 2015, Final 2018, J. Kelliher]
\end{Prob}

\subsection*{Sobolev spaces}

\begin{Prob}
 Fix $d\geq 1$, let $U$ be an open subset of $\R^d$, and let $p\in [1,\infty]$. For any $u\in W^{1,p}(\R^d)$ let $R(u) = u_{|_U}$. Show that this defines a continuous linear operator from $W^{1,p}(\R^d)$ to $W^{1,p}(U)$.
 \vskip 1em
 [Source: Final 2018, J. Kelliher]
\end{Prob}

\begin{Prob}
 Let $U\subseteq \R^d$ be open.
 \begin{enumerate}
  \item [(a)]Integrate by parts to prove the interpolation inequality, 
  $$\|\nabla u\|_{L^2} \leq \|u\|_{L^2}^{1/2} \|\Delta u\|_{L^2}^{1/2}$$
  for all $u\in \mathcal{C}^\infty_c (U)$.
  \item [(b)] Assume now that $U$ is bounded with smooth boundary. Prove that the same inequality as in $(a)$ holds for all $u\in H^2(U) \cap H_0^1(U)$.
 \end{enumerate}
\vskip 1em
[Source: Final 2018, J. Kelliher]
\end{Prob}

\begin{Prob}
Let $U$ be a bounded, open set in $\R^n$ with smooth boundary. Show that
$$\|Du\|^2_{L^2(U)} \leq C\|u\|_{L^2(U)} \|D^2u\|_{L^2(U)} \text{ for all }u\in H^2_0 (U)$$
where $C$ does not depend on $u$.
\vskip 1em
[Source: Midterm 2015, J. Kelliher]
\end{Prob}

\begin{Prob}
 Prove or disprove the following:\vskip 1em
 Let $U$ be a bounded, open subset of $\R^2$. If $u \in W^{1,2}(U)$, then $u\in L^\infty (U)$ with the estimate
 $$\|u\|_{L^\infty (U)} \leq C\|u\|_{W^{1,2}(U)}$$
 where $C$ does not depend on $u$.
 \vskip 1em
 [Source: Midterm 2015, J. Kelliher]
\end{Prob}

\begin{Prob}
 Let $U$ be the open unit ball in $\R^d$.
 \begin{enumerate}
  \item [(a)] Let 
  $$u(x) = |x|^{-\alpha}.$$
  Determine which values of $\alpha >0$, $d\geq 1$, and $p>1$ give $u$ in $W^{1,p}(U)$.
  \item [(b)] Show that
  $$u(x) = \log \log \left( 1+|x|^{-1}\right)$$
  belongs to $W^{1,2}(U)$ but does not belong to $L^\infty (U)$. 
 \end{enumerate}
 
 \vskip 1em
 [Source: Qual 2016, J. Kelliher]
 
\end{Prob}

\begin{Prob}
\begin{enumerate}
 \item [(a)]  Let $u$ be a function on $\R^d$. For any $h\in \R^d$, define $\tau_h u = u(\cdot -h)$: this is translation of $u$ by $h$. Show that 
 
 $$\|\tau_h u -u\|_{L^p} \leq |h| \|\nabla u\|_{L^p}$$
 
 for any $1\leq p <\infty$. 
 
 {\bf Suggestion:} For $1\leq p <\infty$, use the fundamental theorem of calculus trick/technique.
\item[(b)] Now assume that $p\in (d,\infty]$. Take as a given the following lemma:

{\bf Lemma:}
 For all $p\in (d,\infty]$,
 
 $$\|u\|_\infty \leq C_1 \|u\|_{L^p} + C_2\|\nabla u\|_{L^p},$$
 where $C_1$ and $C_2$ depend on $p$ and $d$.

\vskip 1em
Use this lemma to show that for all $u\in W^{1,p}(\R^d)$,
$$\|u\|_{L^\infty} \leq C \|u\|_{L^p}^{1-\theta} \|\nabla u\|_{L^p}^\theta,$$
where $\theta = d/p$.
{\bf Suggestion:} First make a scaling argument to show that 
$$\|u\|_{L^\infty} \leq C_1 \lambda^{\frac{d}{p}}\|u\|_{L^p} + C_2 \lambda^{-1+\frac{d}{p}}\|\nabla u\|_{L^p}.$$
for any $\lambda>0$. (That is, apply the lemma to $u(\cdot /\lambda)$). Then choose the optimal $\lambda$ to obtain the result.

\item[(c)] Conclude from (a) and (b) that for all $u\in W^{1,p}(\R^d)$, $p\in (d,\infty]$,
$$|u(x)-u(y)|\leq C |x-y|^\gamma \|\nabla u\|_{L^p}$$
for $\gamma = 1-d/p$.
\vskip 1em
{\bf Note:} This provides an alternative proof of Morrey's inequality (given a proof of the lemma above).
\end{enumerate}

 
 \vskip 1em
 [Source: Final 2016, J. Kelliher]
\end{Prob}



\subsection*{Others}

\begin{Prob}
 Using separation of variables, solve the wave equation with homogeneous Dirichlet boundary conditions on $(t,x)\in \R\times (0,L)$,
 
 $$\begin{cases}
    \partial_{tt}u = c^2\partial_{xx}u,\\
    u(t,0) = u(t,L) = 0,\\
    u(0,x) = f(x),\\
    \partial_t u(0,x) = g(x).
   \end{cases}
$$
 
 You can assume that $f,g\in \mathcal{C}^\infty ((0,L))$.
 \vskip 1em
 [Source: Final 2018, J. Kelliher]
\end{Prob}

\begin{Prob}
 Consider the one-dimensional conservation law, 
 $$\begin{cases}
    \partial_t u + \partial_x F(u) = 0,\\
    u(0) = g,
   \end{cases}
$$
with initial data
$$g(x) = \begin{cases}
          u_l & \text{if }x<0,\\
          u_r & \text{if }x>0.
         \end{cases}
$$
This is called the \emph{Riemann problem}. The solution is to be on $(0,\infty)\times \R$ (time $\times$ space). We assume that $F$ is smooth and uniformly strictly convex (that is, $F^{\prime \prime}>c>0$ for some constant $c$) and that $u_l$ and $u_r$ are constants.
\begin{enumerate}
 \item [(a)] Define what it means for $u$ to be an entropy solution to the one-dimensional conservation law.
 \item [(b)] Show that if $u_l>u_r$ then 
 $$u(t,x) = \begin{cases}
             u_l & \text{if }\frac{x}{t}<\sigma,\\
             u_r & \text{if }\frac{x}{t}>\sigma
            \end{cases}
$$
is an entropy solution to Riemann's problem, where
$$\sigma:= \frac{F(u_l)-F(u_r)}{u_l-u_r}.$$
\end{enumerate}

 \vskip 1em
 [Source: Qual 2015, J. Kelliher]
\end{Prob}

\section*{207C material}

\subsection*{Maximum Principles for elliptic and parabolic operators}

\begin{Prob}
 Suppose $u$ is a smooth solution to
 
 $$\begin{cases}
    \Delta u = u^3+u & \text{in } U\\
    u=0 & \text{on }\partial U
   \end{cases}
$$

Show that $u\equiv 0$ in $U$ using maximum principle.
\vskip 1em
[Source: Final 2018, P.N. Chen]
\end{Prob}

\begin{Prob}
 Prove that there is at most one smooth solution of
 
 $$\begin{cases}
    u_t - \Delta u = f & \text{in } U_T\\
    \frac{\partial u}{\partial \nu} = 0 & \text{on }\partial U \times [0, T]\\
    u=g & \text{on } U \times \{ 0\}
   \end{cases}
$$
\vskip 1em
[Source: Final 2018, HW, P.N. Chen, Evans' Ch 7]
\end{Prob}

\begin{Prob}
 Suppose $u$ is a smooth solution to 
 
 $$\begin{cases}
    u_t - \Delta u + cu = 0\\
    u=0\\
    u=g
   \end{cases}
$$

where the function $c$ satisfies $c\geq \delta >0$. Show that $|u(x,t)| \leq Ce^{-\gamma t}$.
\vskip 1em
[Source: Final 2018, HW, P.N. Chen, Evans' Ch 7]
\end{Prob}

\begin{Prob}
 Assume that $u$ is a smooth solution of the PDE from the problem above, that $g\geq 0$, and that $c$ is bounded (but not necessarily nonnegative). Show $u\geq 0$. 
 \vskip 0.5em
 {\bf Hint:} What PDE does $v:=e^{-\lambda t}u$ solve?
 \vskip 1em
 [Source: P.N. Chen's HW, Evans' Ch7]
\end{Prob}

\begin{Prob}
Assume $U$ is connected. Use (a) energy methods and (b) the maximum principle to show that the only smooth solutions of the Neumann boundary-value problem 
\[\begin{cases}
   -\Delta u = 0 & \text{in }U\\
   \frac{\partial u}{\partial \nu} = 0 & \text{on }\partial U
  \end{cases}
\]
are $u\equiv C$ for some constant $C$.
\vskip 1em
[Source: P.N. Chen's HW, Evan's Ch 6]
\end{Prob}



\begin{Prob}
We say that the uniformly elliptic operator
$$Lu = -\sum_{i,j=1}^n a^{ij}u_{x_ix_j} + \sum_{i=1}^n b^iu_{x_i} + cu,$$
satisfies the weak maximum principle if for all $u\in \mathcal{C}^2(U) \cap \mathcal{C}(\bar{U})$, 

$$\begin{cases}
    Lu\leq 0 & \text{in }U,\\
    u\leq 0 & \text{on }\partial U, 
  \end{cases}
$$
implies that $u\leq 0$ on $U$.

Suppose that there exists a function $v\in \mathcal{C}^2(U) \cap \mathcal{C}(\bar{U})$ such that $Lv\geq 0$ in $U$ and $v>0$ on $\bar{U}$. Show that $L$ satisfies the weak maximum principle.

{\bf Hint:} Find an elliptic operator $M$ such that $w= y/v$ satisfies $Mw\leq 0$ in the region $\{u>0\}$. To do this, first compute $(v^2 w_{x_i})_{x_j}$.

\vskip 1em
 [Source: Qual 2016, J. Kelliher, P.N. Chen's HW, Evan's Ch6]
\end{Prob}

\begin{Prob}
Let $Lu$ be defined as:

$$Lu = -\sum_{i,j=1}^n a^{ij}(x,t) u_{x_i x_j} + \sum_{i=1}^n b^i (x,t) u_{x_i} + c(x,t) u.$$

Suppose $u\in \mathcal{C}_1^2 (U_T)\cap \mathcal{C}^0 (\bar{U_T})$ satisfies 

$$u_t + Lu \geq 0, \text{  in } U_T.$$

Assume $c(x,t) \geq m$ with $m$ being a fixed constant (either positive or negative), show that if $u\geq 0$ on $\Gamma_T$, then $u\geq 0$ in $\bar{U_T}$.

 \vskip 1em
 [Source: Qual 2016, J. Kelliher]
\end{Prob}


\subsection*{Sobolev Spaces}

\begin{Prob}
 Consider a function $u:(-1,1)\rightarrow \R$ given by $u(x) = |x|$.
 
 \begin{enumerate}
  \item [a)] Show that $u$ is Lipschitz continuous but not continuously differentiable.
  \item [b)] Show that $u$ is weakly differentiable and find the weak derivative.
  \item [c)] How about the second weak derivative? Find the distributional derivative and weak derivative of $u^\prime$ if possible; if not, explain why.
 \end{enumerate}

 \vskip 1em
 [Source: Qual 2015, J. Zhang or J. Kelliher]
\end{Prob}

\begin{Prob}
 Verify thatt if $n>1$, the unbounded function $u = \log \log \left( 1+\frac{1}{|x|}\right)$ belongs to $W^{1,n}(U)$, for $U=B^0(0,1)$.
 \vskip 1em
 [Source: Po-Ning's HW, Evans' Ch.5]
\end{Prob}

\begin{Prob}
 Fix $\alpha >0$ and let $U = B^0(0,1)$. Show there exists a constant $C$, depending only on $n$ and $\alpha$, such that 
 $$ \int_U u^2 dx \leq C \int_U |Du|^2dx,$$
 provided
 \[|\{ x\in U \colon u(x) = 0\}| \geq \alpha,\ u\in H^1(U)\]
 \vskip 1em
 [Source: Po-Ning's HW, Evans' Ch.5]
\end{Prob}

\begin{Prob}
 (Variant of Hardy's inequality) Show that for each $n\geq 3$ there exists a constant $C$ so that
 \[\int_{\R^n} \frac{u^2}{|x|^2} dx \leq C \int_{\R^n} |Du|^2 dx\]
 for all $u\in H^1 (\R^n)$.
 \vskip 0.5 em
 {\bf Hint:} $|Du + \lambda \frac{x}{|x|^2} u|^2\geq 0$ for each $\lambda \in \R$.
 \vskip 1em
 [Source: Po-Ning's HW, Evans' Ch.5]
\end{Prob}

\subsection*{Existence of solutions}

\begin{Prob}
 Let 
 \[Lu = -\sum_{i,j=1}^n (a^{ij}u_{x_i})_{x_j} +cu.\]
Prove that there exists a constant $\mu>0$ such that the corresponding bilinear form $B[\cdot, \cdot]$ satisfies the hypothesis of the Lax-Milgram Theorem, provided
\[c(x) \geq -\mu \ (x\in U).\]
\vskip 1em
[Source: P.N. Chen's HW, Evan's Ch 6]
\end{Prob}

\begin{Prob}
A function $u\in H^2_0 (U)$ is a weak solution of this boundary-value problem for the biharmonic equation

\[\begin{cases}
   \Delta^2 u = f & \text{in }U\\
   u=\frac{\partial u}{\partial \nu} = 0 & \text{on }\partial U
  \end{cases}
\]

provided 

\[\int_U \Delta u \Delta v dx = \int_U fv dx\]

for all $v\in H_0^2 (U)$. Given $f\in L^2(U)$, prove that there exists a unique weak solution of this problem.

\vskip 1em
[Source: P.N. Chen's HW, Evan's Ch 6]
\end{Prob}

\begin{Prob}
 Let $U$ be a connected, bounded open set in $\R^n$. We say that $u\in H^1(U)$ is weak solution of 
 
 $$\begin{cases}
    -\Delta u = f & \text{in }U \\
    \frac{\partial u}{\partial \nu} = 0 & \text{on }\partial U
   \end{cases}
$$
if

$$\int_U Du\cdot Dv dx = \int_U fv dx \text{ for all }v\in H^1(U).$$

Suppose $f\in L^2(U)$. Prove that there exists a weak solution to the problem above if and only if $\int_U f dx = 0$.
\vskip 1em
[Source: Qual 2015, J. Zhang or J. Kelliher, P.N. Chen's HW, Evan's Ch 6]
\end{Prob}

\begin{Prob}
 Consider the following Dirichlet problem
 
 $$\begin{cases}
    -\Delta u + \mu u = f & \text{in }U\\
    u=0 & \text{on }\partial U
   \end{cases}
$$
where $\mu$ is a given nonzero constant.

\begin{enumerate}
 \item [a)] Show the existence of a weak solution $u\in H^1_0(U)$ of the problem above for $\mu >0$.
 \item [b)] Discuss the problem when $\mu <0$.
\end{enumerate}
\vskip 1em
[Source: Qual 2015, J. Zhang or J. Kelliher]
\end{Prob}


\begin{Prob}
Assume $U$ is connected. Consider the Poisson's equation with the Neumann boundary-value problem

$$\begin{cases}
   -\Delta u = f, & \text{on }U,\\
   \frac{\partial u}{\partial n} = 0, & \text{on }\partial U.
  \end{cases}
$$
\begin{enumerate}
 \item [a)] State the definition of the weak solution to this problem; 
 \item [b)] Show that for $f\in L^2(U)$, the above problem has a weak solution if and only if $\int_U f dx = 0$;
 \item [c)] If $f=0$, use the energy method to show that the only smooth solutions of this problems are $u=C$, for some constant $C$.
\end{enumerate}

\vskip 1em
 [Source: Qual 2016, J. Kelliher]
\end{Prob}

\subsection*{Others}

\begin{Prob}
Consider Laplace's equation with potential function $c$:

\begin{equation}\label{eq1}
 -\Delta u +cu = 0,
\end{equation}

and the divergence structure equation:

\begin{equation}\label{eq2}
 -\rm{div}(aDv) = 0
\end{equation}
where the function $a$ is positive. 

\begin{enumerate}
 \item [a)] Show that if $u$ solves (\ref{eq1}) and $w>0$ also solves (\ref{eq1}), then $v:= u/w$ solves (\ref{eq2})  for $a:= w^2$.
 \item [b)] Conversely, show that if $v$ solves (\ref{eq2}), then $u := va^{1/2}$ solves (\ref{eq1}) for some potential $c$.
\end{enumerate}
\vskip 1em
[Source: P.N. Chen's HW, Evan's Ch 6]
\end{Prob}

\end{document}