Difference between revisions of "Aufgaben:Problem 15"

Revision as of 10:25, 10 June 2015

Let \( f \) be a continuous real-valued function on \( \mathbb{R}^n \). Let \(B = B_1(0) \) denote the unit ball in \( \mathbb{R}^3 \) . We define

$$ 1_B= \begin{cases} 1 \space x \in B \\ 0\space x\notin B \end{cases} $$

and a linear map

\begin{split} f \cdot \mu_{\partial B} : \mathcal D (\mathbb{R}^3) &\rightarrow \mathbb{R}^3 \\ \phi &\mapsto (f \cdot \mu_{\partial B})(\phi) = \int_{\partial B} f(x') \phi(x') dx' \end{split}

where \( dx' \) is the surface volume element on \( \partial B \). In particular, \(1 \cdot \mu_{\partial B} =: \mu_{\partial B} \)

a) Show that \(\forall \phi \in \mathcal D (\mathbb{R}^3) \)

\begin{split} |\langle 1_B,\phi \rangle| &\le |B| {||\phi||}_{L^\infty(\mathbb{R}^3)} \qquad \qquad \quad &(1) \\ |\langle f \cdot 1_B,\phi \rangle| &\le |B| {||f||}_{L^\infty(B)}{||\phi||}_{L^\infty(\mathbb{R}^3)} &(2) \\ |\mu_{\partial B}(\phi)| &\le |\partial B| {||\phi||}_{L^\infty(\mathbb{R}^3)} &(3) \\ | (f \cdot \mu_{\partial B})(\phi )| &\le |\partial B| {||f||}_{L^\infty(B)}{||\phi||}_{L^\infty(\mathbb{R}^3)} &(4) \end{split}

b) Prove that \(1_B, f \cdot 1_B, \mu_{\partial B}\) and \( f \cdot \mu_{\partial B}\) are distributions on \( \mathbb{R}^3 \).

Solution of part a)

First we note, that the space of test functions \(\mathcal D (\mathbb{R}^3) \subset \mathcal S (\mathbb{R}^3) \subset L^\infty (\mathbb{R}^3) \) and because \(\phi\) is in the space of testfunctions, it follows that \( \phi \) is integrable.

The extreme value theorem (Weierstrass) states, that a real valued function which is continuous on a compact subset of \( \mathbb{R}^n \) is bounded and takes its minimum and maximum in this subset. This will be used when talking about the function \( f \). It follows that \( f \) is bounded on the unit ball.

(1)

$$|\langle 1_B,\phi \rangle| = |\int_{\mathbb{R}^3} 1_B(x)\phi(x)dx| = |\int_{B}\phi(x)dx| \le | |B| \sup_{x \in B}(\phi (x))| = |B| |\sup_{x \in B}(\phi (x))| \le |B| \sup_{x \in B}|\phi (x)| \le |B| \sup_{x \in \mathbb{R}^3}|\phi (x)| = |B|{||\phi||}_{L^\infty(\mathbb{R}^3)}$$

(2)

$$|\langle f \cdot 1_B,\phi \rangle| = |\int_{\mathbb{R}^3} (f \cdot 1_B)(x)\phi(x)dx| = |\int_{B} f(x)\phi(x)dx| \le | |B| \sup_{x \in B}(f(x)\phi(x))| \le |B| |\sup_{x \in B}(f(x))||\sup_{x \in B}(\phi (x))| \le $$ $$|B| \sup_{x \in B}|f(x)|\sup_{x \in B}|\phi (x)| \le |B| \sup_{x \in B}|f(x)|\sup_{x \in \mathbb{R}^3}|\phi (x)| = |B| {||f||}_{L^\infty(B)}{||\phi||}_{L^\infty(\mathbb{R}^3)} $$

(3)

|\mu_{\partial B}(\phi)| = |(1 \cdot \mu_{\partial B}(\phi)|

\le |\partial B| {||\phi||}_{L^\infty(\mathbb{R}^3)}