Exam
6420-2024-08
Fall 2024 · 6 questions in this exam.
Questions in This Exam
Ordered by the question sequence in the original exam.
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Question 1
State (carefully) and prove the standard \(L^2\) Poincaré inequality in a smooth bounded domain \(U\).
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Question 2
Let \(U\) be a smooth bounded domain and \(f:\overline {U}\to \mathbb {R}\) be a smooth function. Consider the energy
\[ I[v]:=\int _U \left (\sqrt {1+|\nabla v|^2}-fv\right )\,dx \]on the admissible class
\[ \mathcal {A}=\{v\in C^2(U)\cap C^1(\overline {U}): v=0\text { on }\partial U\}. \]Suppose that there is \(u\in \mathcal {A}\) satisfying
\[ I[u]=\min _{v\in \mathcal {A}}I[v]. \]Find and justify the PDE boundary value problem satisfied by \(u\).
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Question 3
Let \(U\) a bounded domain in \(\mathbb {R}^n\). Consider a smooth solution \(u\) of the equation \[ \begin {cases} u_t=\Delta u+c(x,t)u & \text {in } U\times (0,\infty ),\\ u=0 & \text {on } \partial U\times [0,\infty ),\\ u(x,0)=f(x). \end {cases} \]
Show that, if \(c(x, t) \le -\gamma \) for all \(x\) and \(t\), then
\[ \max _{x\in U}|u(x,t)|\le e^{-\gamma t}\max _U |f|. \] -
Question 4
Consider the scalar conservation law,
\[ u_t+\left (\frac 14 u^4\right )_x=0\quad \text {in } \mathbb {R}\times (0,\infty ). \]Find the entropy satisfying weak solution for the initial data, \[ u(x,0)= \begin {cases} 0, & x<0\text { or }x>1,\\ 1, & 0<x<1. \end {cases} \]
Make a drawing of the characteristics in the \((x, t)\)-plane including any shocks and rarefactions.
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Question 5
Use the method of characteristics to solve
\[ xu_x+yu_y=u\quad \text {in } \{y>0\} \]with
\[ u(x,1)=\frac {1}{1+x^2}. \] -
Question 6
Consider the wave equation in a two-dimensional quadrant
\[ u_{tt}-\Delta u=0\quad \text {in } \{x>0,\ y>0\}\times (0,\infty ) \]with boundary conditions \(\partial _x u(0,y,t)=0\) and \(u(x,0,t)=0\) and initial condition \(u_0\), supported in the ball
\[ B_1((2,2))=\{(x,y):(x-2)^2+(y-2)^2\le 1\}. \]Describe, precisely and with careful justification, the support of \(u(x, y, t)\) at all positive times \(t > 0\). Note: The support of a function \(f(x,y)\) is the closure of the set \(\{(x,y): f(x,y)\ne 0\}\).