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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2004 • Question 7

Question

Math 6410 • ODE Fall 2004 • Question 7

Variational Principles. Suppose \(\Omega \in \mathbb {R}^n\) is a smooth, bounded domain. Show that there is an \(\epsilon >0\) depending on \(n\) and \(\Omega \) such that if \(f,g\in C(\overline {\Omega })\) are continuous functions such that \(f\le \epsilon \) then the equation \[ \begin {aligned} \Delta u+f(x)u &= g(x),\quad \text {for }x\in \Omega ,\\ u(x) &= 0,\quad \text {for }x\in \partial \Omega , \end {aligned} \] has a weak solution \(u\in H^1_0(\Omega )\).

Raw LaTeX

\textbf{Variational Principles.}
Suppose \(\Omega\in\R^n\) is a smooth, bounded domain. Show that there is an
\(\epsilon>0\) depending on \(n\) and \(\Omega\) such that if \(f,g\in C(\overline{\Omega})\) are
continuous functions such that \(f\le\epsilon\) then the equation
\[
\begin{aligned}
  \Delta u+f(x)u &= g(x),\quad \text{for }x\in\Omega,\\
  u(x) &= 0,\quad \text{for }x\in\partial\Omega,
\end{aligned}
\]
has a weak solution \(u\in H^1_0(\Omega)\).

Topic Tags

Distributions and Weak Solutions Elliptic Equations Fundamental Solutions Sobolev Spaces and Energy Estimates