Thread: Homogeneous distributions & the Laplacian

I am studying convolution of distributions to solve linear PDEs like the Laplace and Heat equations. A few justifications are skipped in my lecture notes and I'm trying to fill in the gaps:
1. u is homogeneous of degree \lambda ---> {{\partial}^\alpha}u is homog. of degree \lambda - |\alpha|.
2. {{\partial}^\alpha} \delta is homog. of degree -n-|\alpha|

For 1. I know what the definition of a homogeneous function (of degree \lambda) is: f(ax) = a^{\lambda}f(x). Also a distribution u, is homog. (of degree \lambda) if
<u, \phi_a> = 1/(a^n + \lambda) <u, \phi>. I can show that these definitions coincide when u is a function and am trying to use the same method for proving 1. but it doesn't quite work.... For 2. it's easy to show that the delta distribution is homog. of order -n, but that's not what I need to show.....

3. Show |x|^2 \Laplacian \delta = 2n\delta (in \mathbb{R^n}) . I don't have a clue for this one.....Thanks
p.s. I am almost computer illiterate, but does it take this long to fix \Latex.

I'm not completely clear what you're asking here. In 1, is u a distribution or just a function? I also can't follow you're definition of a homogeneous distribution. Why not just say that a distribution f is homogeneous of degree \lambda if f(ax)=a^\lambda f(x)?

How long is Latex going to be down MHF?

Well, that's the definition that we have been given. Of course, distributions are much more generalised than functions and so I guess it makes sense to have a more general definition for homogeneity, which coincides with that of a function when you treat a distribution as a function. Hope this clarifies your question, though I'm still not any closer to answering the questions....

I know what a distribution is. What I'm saying is that your definition is non-standard and in fact looks incorrect. What text are you using for the course?

The definition isn't incorrect. I'm using Friedlander & Joshi's text on distribution theory.

I have solved the problem, so no worries......

Originally Posted by ojones

How long is Latex going to be down MHF?

MathGuru, the owner of MHF, is working on it. In the meantime, there are work-arounds.

Cheers.

The RHS of

<u, \phi_a> = 1/(a^n + \lambda) <u, \phi>

is incorrect.

Also, the notation \phi_a is not standard.

Originally Posted by Ackbeet

MathGuru, the owner of MHF, is working on it. In the meantime, there are work-arounds.

Cheers.

OK, thanks for the update.