- #1
Brian T
- 130
- 31
Homework Statement
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 :)
Regards