Re: inverse of the Laplacian

Graven Water wrote:
David Bernier <david250@xxxxxxxxxxxx> wrote:
For each Euclidean space R^n , n>=2, there's a so-called
fundamental solution to the Laplace equation.

For R^2, there is

f(r) = -1/(2Pi)log( ||r||_2 ) from
< http://www.iam.uni-stuttgart.de/bem/bem_pages/node28.html > .

Suppose the differential form is 1/zbar d zbar, on the complex plane, and it's defined in an annulus around the origin.

If you find a function g with Laplacian = 1/zbar, then for dg/dz, d(dg/dz) / dzbar = 1/zbar.

So dg/dz would look something like log(zbar) + h(z), h analytic in annulus.

But log(zbar) isn't smooth in the annulus. It has a discontinuity.

Does that mean that when you do this convolution with a Green's function to a smooth function, you don't necessarily get something smooth? The integral converges.

Is he wrong in the book about the Dolbeault cohomology H(0,1)(D)=0, for
D an open set in the complex plane? Am I missing something?

(Dolbeault cohomology is cohom. of differential forms wrt d/dzbar)

I'm not familiar with differential forms. However, John Baez has written about them
(including something on Hodge Theory) in:

"This Week's Finds in Mathematical Physics" (Week 182)

< http://math.ucr.edu/home/baez/week182.html > sections 5, 6, 7 and 8

The simplest case of Hodge Theory is the abelian Hodge Theory
from John Baez' part (5) and later.

Some readers may be familiar with Dolbeault's Theorem:
< http://en.wikipedia.org/wiki/Dolbeault_cohomology#Dolbeault.27s_theorem >

Also maybe useful would be the list of speakers and participants at a symposium
on Hodge Theory, because they must understand it:

< http://www.icms.org.uk/archive/meetings/2003/HODGE/sci_prog.html#Speakers >


David Bernier


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