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On page 54 of the Evans for PDE, it states $$\frac{1}{t^n} \iint _{E(t)} \frac{|y|^2}{s^2} \,dy\,ds = \iint_{E(1)}\frac{|y|^2}{s^2} \,dy\,ds = 4$$ saying "We omit the details of this last computation."

I'd love the details though.. I tried integrating myself but I have problems with integrals in $R^n$. Can anyone help me?

Set $E$: $$E(x,t;r):= \left\{ (y,s)\in R^{n+1}\mid s\le t,\Phi(x-y,t-s) \ge \frac{1}{r^n} \right\}$$

Where $\Phi$ is the fundamental solution of the heat equation:

$$\Phi(x,t)=\frac{1}{(4 \pi t)^{n/2}}e^{-\frac{|x|^2}{4t}} \text{ with } t>0.$$

When he states $E(t)$ he means the other two variables, $x$ and $r$, are zero.

The entire proof is here at page 10, http://math.uchicago.edu/~may/REU2014/REUPapers/Ji.pdf , what I'm referring to is at page 11.

Michael Hardy
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    You'll need to tell us what the sets $E(t)$ are. $\qquad$ – Michael Hardy Feb 16 '16 at 16:35
  • You're right, I added the definition along with a pdf with the proof.. What I'm specifically trying to understand is the equation 5.20, in which I think he does the same trick he uses for 5.15. – suspendedbeam Feb 16 '16 at 17:06

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Found the answer How to prove that $\iint\frac{|y|^2}{s^2}\,dy\,ds=4$?

I thought I had searched well enough, seems like I didn't. Sorry for the trouble

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