Skip to content

Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1997 • Question 19

Question

Math 6410 • ODE Fall 1997 • Question 19

Show that for all nonnegative integers \(k\) there is a constant \(c(k,n)<\infty \) so that for all \(u\in H^{k+2}(\mathbb {T}^n)\) we have \[ \norm {u}_{H^{k+2}(\mathbb {T}^n)}\le c\left (\norm {Lu}_{H^k(\mathbb {T}^n)}+\norm {u}_{H^0(\mathbb {T}^n)}\right ). \]

Raw LaTeX

Show that for all nonnegative integers \(k\) there is a constant
\(c(k,n)<\infty\) so that for all \(u\in H^{k+2}(\T^n)\) we have
\[
  \norm{u}_{H^{k+2}(\T^n)}\le
  c\left(\norm{Lu}_{H^k(\T^n)}+\norm{u}_{H^0(\T^n)}\right).
\]

Topic Tags

Sobolev Spaces and Energy Estimates Variation of Constants Formula