Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$, where $\bar u=\dfrac{1}{|\Omega|}\int_{\Omega}u$ and $A&gt;0$. Note that unlike Poincare inequality, the left integral is over $\partial\Omega$ and not $\Omega$. How can I do that?
Suppose $u\in W^{1,1}$ and $\partial u$ is $C^1$. I want to prove the following: $$\int_{\partial\Omega}|u-\bar u|\leq A\int_{\Omega}|\nabla u|$$ where $\bar u=\frac{1}{|\Omega|}\int_{\Omega}u$ and $A&gt;0$. Note that unlike Poincar&eacute; inequality, the left integral is over $\partial\Omega$ and not $\Omega$. How can I do that?  I tried to prove in a similar way to Poincare inequality proof (arguing by contradiction), but with no success. I think I am missing something because of the  $\partial\Omega$ issue.
Jquery validation not working with ckeditor Prove that every terms of a sequence defined by a recurrence relation is a perfect square. I posted what I think is a clever answer to:    But then I noticed the question was posted a while ago on Mar 2.  I wanted to see what the chance of the user logging back in and seeing my answer, but the  says the user was seen on Feb 19.  I thought this was supposed to be the date they last logged in? How could they have asked a question on Mar 2 if they haven't logged in since February?  Thanks Does Sheldon Cooper have Asperger's Syndrome? Cauchy distribution characteristic function I am looking for a quick way to sleep without closing the lid, or using the mouse.  If it matters, I have Lion version OS and a MacBook Air. En Dashes & Hyphens in Same Phrase Without recasting and in lieu of unsightly suspensive hyphenation, could these work? The en dash (which means "to") is used between the figures, and the hyphens are used in the compound modifier. To me, the examples below are concise and clear.  Do you agree that the examples below are 100% punctuated correctly--yes or no?  45–50-year-old men  10–20-mile radius  $45–$55-million-per-year industry  $100,000–$150,000-a-year savings  10–15-degree temperature difference  $2–$5-per-day service fees  45–75-cent-a-week raises  55–65-mph speed zone  15–20-ounce steaks  20%–40%-a-year tax increases Why does my hard drive LED light blink every second? The light blinks faintly about once a second.   It is slowly driving me insane.  How do I find out which program is causing this, so I can disable it?  Acer 5274 laptop/core i3 Windows 7 Home Premium Hitachi HTS545050B9A300 hard drive Are there examples of third-(or higher)-order linear differential equations in physics or applied mathematics? The classical second-order linear ordinary differential equation is that named after Sturm and Liouville: formally, \begin{equation} (pu')'=ru. \end{equation} It arises naturally in many physical situations, for example through radial considerations of Schrödinger's equation. A higher-order "analogue" of it is the Orr$-$Sommerfeld equation, which, after relabelling the coefficients, may be written as \begin{equation} (\phi u'')''=\psi u. \end{equation} It also arises naturally, namely from certain simplifications applied to Navier$-$Stokes' (in)famous equation, and describes to great accuracy the cross-stream behaviour of channel fluid flow.  I have been thinking long and hard about whether there are additional higher-order linear differential equations that emerge naturally from our mathematical models of the world. It even seems pretty much all of the linear partial differential equations (heat, wave, Schrödinger, etc.) are of second order. Does anybody know of higher-order examples? Lebesgue density theorem in the line Inertial frames of reference I'm struggling with the notion of an inertial frame of reference. I suspect my difficulty lies with the difference between Newtonian and relativistic inertial frames, but I can't see it.   I've read that Newton's laws apply in any non accelerating frame of reference, which are called inertial frames. So, if I play pool on a train moving with uniform velocity, the balls behave in the same way as if I were playing pool in a pool hall. So the train is (to a good approximation, ignoring forces caused by the Earth's rotation and movement around the Sun etc) an inertial frame.   I've also read that in an inertial frame, Newton's first law is satisfied. So, if I slide a rock on a sheet of ice and I could somehow eliminate the frictional forces between the rock and the ice, the rock would carry on sliding for ever, as predicted by Newton's first law.   Question 1 - is the  sheet of ice also therefore a good approximation to an inertial frame?  Question 2 - are these two definitions of inertial frames saying the same thing in different ways?  Both these examples occur in gravitation fields, which doesn't seem to matter as both the train and the ice sheet are presumably good approximations to inertial frames.  Question 3 - is gravity irrelevant when defining these two inertial frames?  In special relativity, I've read (Foster and Nightingale) that an inertial frame is also one where Newton's first law holds. But as it's special relativity there cannot be an inertial frame if there's a gravitational field, so I'm assuming the above two examples are not inertial frames in special relativity.  Question 4 - how come you can use the same definition of an inertial frame (satisfied Newton's first law) but in the case of special relativity, the train and the ice sheet aren't inertial frames. Is this to do with not being able to synchronize clocks in a gravitational field?  In general relativity, I've read (Schutz) that a freely falling frame is (on Earth) the only possible (and local) inertial frame. So again the first two examples would not be inertial frames in general relativity.  Question 5 - am I right to think that my two examples aren't even approximate approximations to an inertial frame in relativity?  Question 6 - have I missed anything else that might be useful?  Many thanks  Edit. On reflection, am I right in thinking that because Newtonian mechanics assumes universal, absolute time we don't need to worry about synchronizing clocks in a Newtonian inertial frame. Therefore we don't need to worry about gravity in a Newtonian inertial frame, because in such a frame gravity does not affect time.  This is not the case in spacetime, as here gravity does affect time and the only way to have synchronized clocks in an inertial frame in spacetime is: 1. have no gravitational field, or 2. use a local, freely falling frame.  Am I on the right track here? Stack Exchange OpenID credentials not removed on logout, allowing insecure re-login In a photoelectric effect we remove electrons from a metal using high energy photons, the work function is the minimal energy required for this effect.  My question is why doesnt the work fucntion depended on time? The more electrons we take out from the metal, the more ionized it becomes (if we remove a lot of negative charge, it will turn positive), and as a result we would expect to need more energy for this effect (the possitive charge of the metal would make it harder to remove the remaining electrons).  Why does the work function remains constant no matter how much electrons we take out? Counting surjective functions Disk space usage doesn't add up with df & du Error: Declaration of MyClass::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() I'm currently working on a WordPress site and whenever in the admin I go the Appearance > Menus page I get the following error:     ErrorException: Runtime Notice: Declaration of Walker_Nav_Menu_Edit::start_lvl() should be compatible with that of Walker_Nav_Menu::start_lvl() in wp-admin/includes/nav-menu.php line 203   It seems like some sort of PHP compatibility issue. I'm running PHP 5.3.10 on my local dev machine.  What would be the best way to remedy this problem? Consider a stochastic process $\{X_t, t = 1, 2, \ldots\}$ following the model $$X_t = \alpha X_{t-1} + e_t,$$ where $e_t \thicksim f$.  Can I say that the distribution of the initial point, $X_1$, is the same as $f$? Can I say that the stationary marginal density, if it exists, of $\{X_t\}$ is same as $X_2 (\stackrel{D}{=}\alpha X_1 + e_2)$?  I think that the stationary marginal density, if it exists, of $\{X_t\}$ is the same as $X_2$, but not necessarily same as $X_1$. Why is serial data transmission faster than parallel? Intuitively, you would think that parallel data transmission should be faster than serial data transmission; in parallel you are transferring many bits at the same time, whereas in serial you are doing one bit at a time.  So what makes SATA interfaces faster than PATA, PCI-e devices faster than PCI, and serial ports faster than parallel? What is the difference between sourcing ('.' or 'source') and executing a file in bash? What's the difference between executing a script like this:  ./test.sh  and executing a script like this:  . test.sh?  I tried a simple, two-line script to see if I could find if there was a difference:  #!/bin/bash ls   But both . test.sh and ./test.sh returned the same information. Possible Duplicate:          The square root of a number squared is equal to the absolute value of that number. Why is $\sqrt{x^2} = |x|$? Why not just $x$? Please give me a reason and also help me prove it.