The characteristic (indicator) function of a set is not in the Sobolev space H1 - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T17:19:59Z http://mathoverflow.net/feeds/question/25154 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25154/the-characteristic-indicator-function-of-a-set-is-not-in-the-sobolev-space-h1 The characteristic (indicator) function of a set is not in the Sobolev space H1 Spencer 2010-05-18T16:34:39Z 2010-08-26T06:12:47Z <blockquote> <p>Is it true that the characteristic (indicator) function of a subset of Euclidean space with finite positive measure is never in the Sobolev space $H^1 = W^{1,2}$ And if so, what is the best/easiest/most elementary way to see this?</p> </blockquote> <p>Context:</p> <p>I have this on good authority (it is stated in a decent textbook). However, I have had no joy in showing this to be the case myself. Due to its placement in aforementioned textbook as the first exercise at the end of a chapter about $H^1$, it feels like it oughtn't be difficult to show, but a group of my friends and I had no luck. It is bugging me now.</p> <p>[Though I am a student taking a course based on the textbook, this is not `homework'; I will not be graded on it in any way and I have attempted the problem myself.] </p> http://mathoverflow.net/questions/25154/the-characteristic-indicator-function-of-a-set-is-not-in-the-sobolev-space-h1/25156#25156 Answer by Tom Boardman for The characteristic (indicator) function of a set is not in the Sobolev space H1 Tom Boardman 2010-05-18T17:04:40Z 2010-05-18T17:10:14Z <p>If it's in $H^1$ it's a.e. differentiable, with weak differential a.e. equal its differential, which for an indicator function is a.e. zero. So if you integrate any candidate for your weak derivative multiplied by a compactly supported test function you should get zero. Now if you use the right test function and the definition of a weak derivative you ought to be able to contrive a contadiction of the form "$0=1$".</p> http://mathoverflow.net/questions/25154/the-characteristic-indicator-function-of-a-set-is-not-in-the-sobolev-space-h1/25158#25158 Answer by Pietro Majer for The characteristic (indicator) function of a set is not in the Sobolev space H1 Pietro Majer 2010-05-18T17:18:37Z 2010-05-18T18:39:50Z <p>One reason is this: if $f$ is in $H^1({\mathbb R^n})$, you have $\int_{\mathbb R^n}|f(x+h)-f(x)|^2 dx\le C|h|^2$ for all $h \in{\mathbb R^n}$ . Now in the case of $f:=\chi_E$ the integral is just the $L^1$ distance, $\|\chi_E - \chi_{E-h}\|_1$. By the triangular inequality, for any positive integer m and any h as above one gets</p> <p>$\|\chi_E -\chi_{E-h}\|_1 \le$ </p> <p>$\sum_{j=0}^{m-1}\|\chi_{E - \frac{j}{m} h } - \chi_{E-\frac {j+1}{m}h}\|_1 \le$ </p> <p>$Cm|h/m|^2=C|h|^2/m$, </p> <p>whence $\|\chi_E - \chi_{E-h}\|_1=0$ for all $h$. This is impossible since for $|h|\to\infty$ it converges to 2meas(E).</p> http://mathoverflow.net/questions/25154/the-characteristic-indicator-function-of-a-set-is-not-in-the-sobolev-space-h1/36707#36707 Answer by Dorian for The characteristic (indicator) function of a set is not in the Sobolev space H1 Dorian 2010-08-26T00:44:14Z 2010-08-26T06:12:47Z <p>The answer posted by Tom, as written is actually not true. A function in $H^1$ will not in general be differential almost everywhere; it depends on the dimension. In one dimension however it is indeed true that $H^1$ functions are differentiable almost everywhere (they are in fact absolutely continuous). There are two ways of seeing it is not in $H^1$. The simple answer is that if you differentiate the characteristic function of say $[0,\infty)$ then you will get the dirac measure. However let me just answer your question first:</p> <p><strong>Answer 1:</strong> Take any smooth compactly supported $\phi:\mathbb{R} \to \mathbb{R}$. By definition of weak derivative we have $\int \phi g^{\prime} dx = - \int \phi^{\prime} g dx$ where I've set $g=1_{[0,\infty)}$. This would have to be true for all such $\phi$ if the weak derivative existed. Now take $\phi^{\epsilon}$ to be supported in a neighborhood $(-\epsilon,\epsilon)$ of $0$. We are making the crucial assumption that $g^{\prime}$ is an <strong>integrable</strong> and hence it follows that $\int \phi^{\epsilon} g^{\prime} \to 0$ as $\epsilon \to 0$. However, $\phi^{\epsilon}$ is smooth and so $\int \partial_x\phi^{\epsilon}(x)g(x)dx = \phi^{\epsilon}(0)$ since $\phi$ was assumed to have compact support in $(-\epsilon,\epsilon)$. Now just fix $\phi^{\epsilon}(0)=1$ and we have that $\phi^{\epsilon}(0) \to 0$ by the first integral equality. This is a clear contradiction.</p> <p>Notice that in fact that this really shows that $g' dx = \delta(x)$.</p> <p><strong>Answer 2:</strong> Take $1_{[0,1]}$ instead so that it is an $L^2([0,1])$ function. This is in fact the fourier transform of a "sinc" function, $\sin(k)/k$ up to some normalization constants. If we consider the $H^1$ norm in frequency space we would need $\int_0^{\infty} |k|^2\frac{\sin(k)^2}{|k|^2} &lt; \infty$ which is clearly false. This requires being at ease with the fourier transform so if you're not, answer 1 is probably best.</p> <p>It is true in $\mathbb{R}^n$ that if $u \in W^{1,p}$ for $p > n$ then $u$ is a.e differentiable and equals a.e its weak gradient (see Evans chapter 5). This is to correct what Tom had said although perhaps we was thinking about the $n=1$ case in which case $2 > 1$.</p> <p>Hope this helps! Dorian</p> http://mathoverflow.net/questions/25154/the-characteristic-indicator-function-of-a-set-is-not-in-the-sobolev-space-h1/36709#36709 Answer by Nate Eldredge for The characteristic (indicator) function of a set is not in the Sobolev space H1 Nate Eldredge 2010-08-26T01:15:05Z 2010-08-26T01:15:05Z <p>A not-so-elementary way to see it is to use the theorem that if $f$ is an $H^1$ function, then for almost every line segment, the restriction of $f$ to that line segment gives an absolutely continuous function. (In fact, this is essentially sufficient as well as necessary.) Wikipedia cites Maz'ya's book <em>Sobolev spaces</em>, and I also found a proof in <a href="http://books.google.com/books?id=2_vUE9VWghsC&amp;lpg=PA48&amp;ots=n8dL_dOzRS&amp;dq=sobolev%20absolutely%20continuous%20lines&amp;pg=PA44#v=onepage&amp;q=sobolev%20absolutely%20continuous%20lines&amp;f=false" rel="nofollow">Ziemer</a>.</p>