Heat equation

Joubert

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Mar 1, 2011, 5:19:25 PM3/1/11

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Hi everyone.
Suppose f is in L^1 of R^n.
I need to prove that the usual solution of the cauchy problem for the heat
equation, which is given by the convolution of f with the heat kernel is
indeed a solution. In order to do this I need to prove that I can take the
laplacian under the integral sign. Can you help?

Han de Bruijn

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Mar 2, 2011, 3:42:33 AM3/2/11

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Oh yeah, I can help, but don't know it will be appreciated by others
or even by you (because they think I'm one of the cranks here). Fact
is that you can _always_ take the laplacian under the integral sign,
and much more. Here is some of the reason why:

http://hdebruijn.soo.dto.tudelft.nl/jaar2010/limits.txt

Proper limits do always commute.

Han de Bruijn

David C. Ullrich

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Mar 2, 2011, 9:30:19 AM3/2/11

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Since you don't explain what part you're stuck on it's hard to see how
to "help" other than by just writing out a solution for you.

In particular, choose one:

(i) You're unaware of the basic theorem from "reals" that
says that under certain conditions you can differentiate
under the integral sign (this is a simple consequence
of the Mean Value Theorem and Dominated Convergence).

(ii) You're aware of that theorem but don't see how to apply it here.

David C. Ullrich

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Mar 2, 2011, 9:32:25 AM3/2/11

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Giggle. Crank indeed. You can say that "proper limits"
always commute if you want. Unfortunately that's no
help here, because it's a problem about _limits_, not
about "proper limits".

>
>Han de Bruijn

Joubert

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Mar 2, 2011, 2:52:19 PM3/2/11

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Hello David,

> (ii) You're aware of that theorem but don't see how to apply it here.
>

I'm well aware of its existence but couldn't find a "dominating" function.

Han de Bruijn

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Mar 3, 2011, 3:03:21 AM3/3/11

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On Mar 2, 3:32 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Wed, 2 Mar 2011 00:42:33 -0800 (PST), Han de Bruijn
>

Giggle? Improper limits effectively DO NOT EXIST. Fact is that common
mathematics is involved with many limits that effectively do not exist
while they do all that effort for developing all sort of complicated
but nevertheless completely redundant theories about them. Efficiency
is quite another matter than this. In real world, all limits commute.

Han de Bruijn

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Mar 3, 2011, 3:14:42 AM3/3/11

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On Mar 2, 3:30 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
> On Tue, 1 Mar 2011 22:19:25 +0000 (UTC), Joubert
>

> <LVahlf...@nospam.com> wrote:
> >Hi everyone.
> >Suppose f is in L^1 of R^n.
> >I need to prove that the usual solution of the cauchy problem for the heat
> >equation, which is given by the convolution of f with the heat kernel is
> >indeed a solution. In order to do this I need to prove that I can take the
> >laplacian under the integral sign. Can you help?
>
> Since you don't explain what part you're stuck on it's hard to see how
> to "help" other than by just writing out a solution for you.
>
> In particular, choose one:
>
> (i) You're unaware of the basic theorem from "reals" that
> says that under certain conditions you can differentiate
> under the integral sign (this is a simple consequence
> of the Mean Value Theorem and Dominated Convergence).

Under "certain" conditions ? No doubt: under _all_ conditions !

> (ii) You're aware of that theorem but don't see how to apply it here.

One can _always_ exchange integration and differentiation.

Try to deny.

Han de Bruijn

David C. Ullrich

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Mar 3, 2011, 9:21:10 AM3/3/11

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Ok, then remind us what the kernel is...

>

Joubert

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Mar 3, 2011, 9:37:17 AM3/3/11

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The first equation on this page
http://en.wikipedia.org/wiki/Heat_kernel
I guess I can find a dominating function for the derivative of that expression
since it is even bounded for x in R^d