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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Winter 2007 • Question 10

Question

Math 6410 • ODE Winter 2007 • Question 10

Let \(U=\{x\in \mathbb {R}^2:0<\lvert x\rvert <1\}\), and denote \(\Gamma =\{\lvert x\rvert =1\}\). Consider \[ \begin {aligned} -\Delta u &= f, && \text {in } U,\\ u &= 0, && \text {on } \{x=0\},\\ \frac {\partial u}{\partial \eta } &= 0, && \text {on } \Gamma , \end {aligned} \] where \(\eta \) is the unit outward normal, and \(f\in L^2(U)\).

Prove a Poincaré-type inequality for this problem, and use it to show that there exists a unique weak solution \(u\) in an appropriate Hilbert space.

Raw LaTeX

Let \(U=\{x\in\R^2:0<\lvert x\rvert<1\}\), and denote
\(\Gamma=\{\lvert x\rvert=1\}\). Consider
\[
\begin{aligned}
  -\Delta u &= f, && \text{in } U,\\
  u &= 0, && \text{on } \{x=0\},\\
  \frac{\partial u}{\partial\eta} &= 0, && \text{on } \Gamma,
\end{aligned}
\]
where \(\eta\) is the unit outward normal, and \(f\in L^2(U)\).

Prove a Poincar\'e-type inequality for this problem, and use it to show that there
exists a unique weak solution \(u\) in an appropriate Hilbert space.

Topic Tags

Distributions and Weak Solutions Fundamental Solutions Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem