1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

PDE/Analysis question

  1. Feb 16, 2016 #1
    1. The problem statement, all variables and given/known data
    My question involves whether or not we can pass the derivative under the integral in the following question, and what conditions need to be met. My second is question is how to evaluate using Green's identity at the "boundary". See below

    A function u satisfies the diffusion equation in [itex]\mathcal{R}^n \times (0,\infty)[/itex] with some initial value.
    $$u_t = \Delta u, u(x,0)=v(x) $$
    The question is to show that the integral of [itex]u(x,t)[/itex] over the whole domain is constant w.r.t time (physically, this means the total amount of whatever is diffusing remains constant over all space, e.g. mass conservation). We are also given that the value of the solution [itex]u(x,t) \rightarrow 0 \ as \ |x| \rightarrow \infty[/itex], i.e. that it asymptotically approches 0.
    To show that it is constant w.r.t time, we show that
    $$\frac{d}{dt} \int_{\mathcal{R}^n}u(x,t)d^nx = 0$$
    My question is, if we know that the function vanishes at infinity, can we pass the differentiating under the integral. What are the conditions to be able to do this?
    2. Relevant eq
    3. The attempt at a solution

    Assuming that we can pass the derivative under the integral,
    $$\frac{d}{dt} \int_{\mathcal{R}^n}u(x,t)d^nx = \int_{\mathcal{R}^n} u_t(x,t)d^nx = \int_{\mathcal{R}^n} \Delta u(x,t)d^nx$$
    Using Green's identity here, where we take the "boundary" to be a sphere [itex]S^{n-1}_R[/itex] and let the radius R go to infinity.
    $$ = lim_{R \rightarrow \infty} \int_{S^{n-1}_R} \frac{\partial u}{\partial n} d^{n-1}s$$
    and since the function vanishes asymptotically, it's derivative will be 0 at this "boundary" and so the whole term will be 0,
    $$\therefore \frac{d}{dt} \int_{\mathcal{R}^n}u(x,t)d^nx = 0$$

    The questions I have with my solution is
    (1) in what conditions are we allowed to pass the derivative under the integral as I did in my solution. Was this justified? (Although the vanishing at infinity is weaker than the condition for compact support, I treated them essentially as the same. I'd like more details on this)
    (2) Was it justified to use Green's theorem on a "boundary" whose size we let go to infinity?
    If anyone has a more rigorous approach at this problem, please clue me in :)
  2. jcsd
  3. Feb 16, 2016 #2


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    The limit in the definition of the derivative must commute with the integration. As long as it does you will be fine.
    Yes, but you are missing that this is not sufficient. It is generally possible for an integral over a growing domain to have a non zero limit even if the integrand goes to zero.
  4. Feb 16, 2016 #3
    Thanks for the insight. So for your second comment, does this imply that we have to assume that the function has compact support?
  5. Feb 17, 2016 #4


    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member
    2017 Award

    No. Just that you need to argue better. Due to the behaviour of the Green function, it will not have compact support.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted