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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2000 • Question 7

Question

Math 6410 • ODE Fall 2000 • Question 7

Let \(\Omega \) be a bounded open set. Define the Sobolev space \(H^{1,2}_0(\Omega )\). Indicate why this is a Hilbert space. State and prove Poincaré’s inequality for functions \(u\in H^{1,2}_0(\Omega )\) and thus verify that \(H^{1,2}_0(\Omega )\) is also a Hilbert space with respect to the inner product \[ (u,v)=\int _\Omega \nabla u\cdot \overline {\nabla v}\,dx. \]

Raw LaTeX

Let \(\Omega\) be a bounded open set. Define the Sobolev space
\(H^{1,2}_0(\Omega)\). Indicate why this is a Hilbert space. State and prove Poincar\'e's
inequality for functions \(u\in H^{1,2}_0(\Omega)\) and thus verify that
\(H^{1,2}_0(\Omega)\) is also a Hilbert space with respect to the inner product
\[
  (u,v)=\int_\Omega \nabla u\cdot\overline{\nabla v}\,dx.
\]

Topic Tags

Hilbert Spaces and Orthogonality Poincare Maps and Periodic Solutions Poincare-Bendixson Theorem Sobolev Spaces and Energy Estimates