Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6410 • ODE

Fall 2001 • Question 6

Question

Math 6410 • ODE Fall 2001 • Question 6

Let \(G\subset \mathbb {R}^N\) be an open set. Define what is meant by a distribution on \(G\). Also define the concept of distributional derivatives of any order \(\alpha \), where \(\alpha \) is a multiindex.

Give a definition of the Sobolev spaces \(H^m(G)\), \(H^m_0(G)\), where \(m\) is a nonnegative integer. Use properties of distributions to prove the generalized integration by parts formula \[ \inner {\partial ^\alpha f}{g} = (-1)^{\lvert \alpha \rvert }\inner {f}{\partial ^\alpha g}, \qquad f\in H^m(G),\ g\in H^m_0(G),\ \lvert \alpha \rvert \le m, \] and \(\inner {\cdot }{\cdot }\) is the \(L^2(G)\) inner product.

Let \(G\subset\R^N\) be an open set. Define what is meant by a distribution on \(G\). Also
define the concept of distributional derivatives of any order \(\alpha\), where \(\alpha\)
is a multiindex.

Give a definition of the Sobolev spaces \(H^m(G)\), \(H^m_0(G)\), where \(m\) is a
nonnegative integer. Use properties of distributions to prove the generalized
integration by parts formula
\[
  \inner{\partial^\alpha f}{g}
  =
  (-1)^{\lvert\alpha\rvert}\inner{f}{\partial^\alpha g},
  \qquad
  f\in H^m(G),\ g\in H^m_0(G),\ \lvert\alpha\rvert\le m,
\]
and \(\inner{\cdot}{\cdot}\) is the \(L^2(G)\) inner product.

Topic Tags

Distributions and Weak Solutions Hilbert Spaces and Orthogonality Sobolev Spaces and Energy Estimates