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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 16

Question

Math 6410 • ODE Fall 1996 • Question 16

Let \(G\) be a bounded open set. State and prove Poincaré’s inequality for functions \(u\in W_0^{1,2}(G)\) and thus verify that \(W_0^{1,2}(G)\) is also a Hilbert space with respect to the inner product \[ (u,v)_{W_0^{1,2}(G)}=\int _G \nabla u\cdot \overline {\nabla v}\,dx. \]

Raw LaTeX

Let \(G\) be a bounded open set. State and prove Poincar\'e's inequality for
functions \(u\in W_0^{1,2}(G)\) and thus verify that \(W_0^{1,2}(G)\) is also a Hilbert
space with respect to the inner product
\[
(u,v)_{W_0^{1,2}(G)}=\int_G \nabla u\cdot\overline{\nabla v}\,dx.
\]

Topic Tags

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