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Math 6410 • ODE

Fall 2003 • Question 8

Question

Math 6410 • ODE Fall 2003 • Question 8

Distributions. Let \(\Omega \subset \mathbb {R}^n\) be an open set and \(\mathcal {D}'(\Omega )\) the space of distributions on \(\Omega \).

Let \(\delta \in \mathcal {D}'(\mathbb {R}^n)\) be the Dirac delta function. Fix a multiindex \(\beta \). Show that the derivative \(D^\beta \delta \) is a distribution.
Let \(C^\infty _0(\Omega )\) denote the space of compactly supported smooth functions on \(\Omega \). Briefly explain why there exist nonzero \(\varphi \in C^\infty _0(\Omega )\).
Suppose \(\rho \in C^\infty _0(\mathbb {R}^n)\) is such that \(\rho \ge 0\), the support is contained in the unit ball around zero, \(\operatorname {spt}\rho \subset B_1(0)\), and \(\int _{\mathbb {R}^n}\rho \,dx=1\). Let \(\eta _h=h^{-n}\rho (x/h)\). Show that as distributions, \(\eta _h\to \delta \) as \(h\to 0\).

\textbf{Distributions.}
Let \(\Omega\subset\R^n\) be an open set and \(\mathcal{D}'(\Omega)\) the space of
distributions on \(\Omega\).
\begin{enumerate}
\item[(a.)] Let \(\delta\in\mathcal{D}'(\R^n)\) be the Dirac delta function. Fix a
multiindex \(\beta\). Show that the derivative \(D^\beta\delta\) is a distribution.
\item[(b.)] Let \(C^\infty_0(\Omega)\) denote the space of compactly supported smooth
functions on \(\Omega\). Briefly explain why there exist nonzero
\(\varphi\in C^\infty_0(\Omega)\).
\item[(c.)] Suppose \(\rho\in C^\infty_0(\R^n)\) is such that \(\rho\ge0\), the support is
contained in the unit ball around zero, \(\operatorname{spt}\rho\subset B_1(0)\), and
\(\int_{\R^n}\rho\,dx=1\). Let \(\eta_h=h^{-n}\rho(x/h)\). Show that as distributions,
\(\eta_h\to\delta\) as \(h\to0\).
\end{enumerate}

Topic Tags

Distributions and Weak Solutions Fundamental Solutions