## Laplacian

Posted October 18th, 2007 by Isoscel

Laplace equation in two dimension

can be written in polar coordinates with inverse restriction expressed by

let

Then

We have

If we suppose that our function is of class than mixed partial derivative are equal by Young or Schwartz theorem so we can write

(*)

(**)

For a more condensed expression it is useful to introduce

and inner product

Then summing up (*) and (**) we have

We have so

and finally