Like solutions to the Laplace equation, (classical) solutions to the heat equation satisfy a mean value property. But instead of integrals over balls or spheres, the heat mean value property involves integrals over a heat ball. The heat ball of radius centered at a point of in , denoted is the set
where denotes the heat kernel
When , we sometimes write instead of .
After playing around with the inequality , one can verify that
From the above expressions, it is easy to see that is bounded and closed. Furthermore, the “center” actually lies on the boundary, so it is somewhat of a misnomer. I leave it to the reader to verify that and is obtained from by the parabolic scaling .
The following theorem of N.A. Watson establishes the heat ball mean value property.
for all , where equality holds if is a solution of the heat equation.
After seeing the proof of this surprisingly recent theorem, I wondered if there was a corresponding result for the boundary of , a heat sphere mean value property. There was no proof of reference for a proof of such a result in the book I was reading, and Watson’s paper makes no mention of such a result either. I asked a professor at Georgia Tech if he knew of mean value property for the heat sphere. He did not, but he gave me a very helpful suggestion which led to the following result; many thanks to Professor F. Bonetto.
for all .
It appears Theorem 2 is at least folklore, as I found the result contained in some very nice lecture notes of Professor G. Menon.
Interestingly, proving the mean value property for the heat equation is done in the reverse order of with the Laplace equation. With harmonic functions, one shows the mean value property holds over spheres and then uses integration in polar coordinates to show the property holds over balls. In contrast, we will show that the mean value property holds over heat balls and then use this to show that it holds over heat spheres.
To prove Theorem 1, I will follows [Evans].
Proof of Theorem 1: As the translate of is again a solution of the heat equation (on a translated domain), we may assume without loss of generality that . For , define
where we make the change of variable . Since is and is a bounded domain, we can differentiate inside the integral to obtain
We introduce the function
which can be obtained by taking the logarithm of . Observe that on and therefore on . We can write the expression as
where we integrate by parts with respect to and use the fact that vanishes on for the boundary term. Integrating by parts now with respect to , we obtain
Since is a subsolution of the heat equation, we have the inequality
where we integrate by parts the first term (w.r.t. ) to obtain the penultimate expression. Therefore is nondecreasing. Using the dominated convergence theorem and the continuity of , we obtain
The result , as it is an exercise in integration, but I have uploaded a PDF of the computation. Dividing both sides by yields the desired inequality. In particular, if is a solution then equality holds.
Using the preceding result, we now prove Theorem 2. For convenience, we introduce the notation
Proof of Theorem 2: Since the translate of is again a solution of the heat equation, we may assume without loss of generality that . Using polar coordinates, we can write
Differentiating with respect to , we obtain
Note that the second term in the expression above vanishes since . We compute
and note that
If is a (classical) solution to the heat equation, then our proof of the mean value property over the heat ball shows that . Rearranging terms, we obtain
Multiplying both sides of the equality by completes the proof.
- L.C. Evans, Partial Differential Equations (Second Edition), AMS, 2010.
- G. Menov, “Lectures on Partial Differential Equations”, Accessed: 10/11/14.
- N.A. Watson, “A Theory of Subtemperatures in Several Variables”, Proc. Lon. Math. Soc. s3-26 (1973), 385-417.
- X. Yu, “Heat Equation – Maximum Principles”, Accessed: 10/1/14.