Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1992 • Question 6

Question

Math 6410 • ODE Fall 1992 • Question 6

Consider the Dirichlet problem in a bounded plane domain \(\Omega \): \[ u_{xx}+u_{yy}=0\quad \text {in }\Omega , \] \[ u=f\quad \text {on }\partial \Omega , \] where \(f\) is continuous on \(\partial \Omega \).

Prove that this Dirichlet problem has at most one solution. Define clearly what you mean by a solution.
Prove that if \(\Omega \) is a bounded domain for which the Dirichlet problem is solvable for all \(f\in C(\partial \Omega )\) then the mapping \(f\in C(\partial \Omega )\longrightarrow u\in C(\overline {\Omega })\) is continuous in the uniform norms. (\(\overline {\Omega }\) denotes the closure of \(\Omega \).)

DE Qualifying Examination 1992

Consider the Dirichlet problem in a bounded plane domain
\(\Omega\):
\[
u_{xx}+u_{yy}=0\quad\text{in }\Omega,
\]
\[
u=f\quad\text{on }\partial\Omega,
\]
where \(f\) is continuous on \(\partial\Omega\).

\begin{enumerate}
\item[(a)] Prove that this Dirichlet problem has at most one solution. Define clearly
what you mean by a solution.

\newpage

\item[(b)] Prove that if \(\Omega\) is a bounded domain for which the Dirichlet
problem is solvable for all \(f\in C(\partial\Omega)\) then the mapping
\(f\in C(\partial\Omega)\longrightarrow u\in C(\overline{\Omega})\) is continuous in
the uniform norms. (\(\overline{\Omega}\) denotes the closure of \(\Omega\).)
\end{enumerate}

\textit{DE Qualifying Examination 1992}

Topic Tags

Boundary Value Problems Elliptic Equations Gronwall Inequality and Continuous Dependence Poincare Maps and Periodic Solutions