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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 1996 • Question 15

Question

Math 6410 • ODE Fall 1996 • Question 15

Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(W^{m,2}(G)\), and \(W_0^{m,2}(G)\) and show that \(H^m(G)\) is a Hilbert space. Give conditions under which any two of the three spaces are equal.

Raw LaTeX

Let \(G\) be an open set. Define the Sobolev spaces \(H^m(G)\), \(W^{m,2}(G)\), and
\(W_0^{m,2}(G)\) and show that \(H^m(G)\) is a Hilbert space. Give conditions under
which any two of the three spaces are equal.

Topic Tags

Hilbert Spaces and Orthogonality Sobolev Spaces and Energy Estimates