Toronto Math Forum
APM3462016F => APM346Lectures => Chapter 8 => Topic started by: Shaghayegh A on November 21, 2016, 10:58:36 PM

link: http://www.math.toronto.edu/courses/apm346h1/20169/PDEtextbook/Chapter8/S8.P.html (3a)
Since P(x,y,z) is a polynomial of degree 0, it is a constant. So $U=x^2+y^2+z^2c_0 (x^2+y^2+z^2)$, but you can't write this as a sum of homogenous harmonic polynomials since there is a term c_0 remaining?

I think you need to plug in the equation to solve for $C_0$, for general $C_0$ function $u$ may not be harmonic.

Solution is a harmonic polynomial but not necessarily homogeneous. On the other hand it must have prescribed boundary value (as $x^2+y^2+z^2=R^2$).

modified:
My solution is $U=x^2+y^2z^2\frac{1}{3} (x^2+y^2+z^21)$, the laplacian of this equation is zero and it equal g(x,y,z) at the boundary. How can I write this as a sum of harmonic homogenous polynomials, since there is a factor of 1/3 : $U=2/3 x^2+ 2/3 y^2 4/3 z^2+ \frac{1}{3}$
By the way, g is $x^2+y^2z^2$

Laplacian is not $0$, and what $g$ you are looking for?

Oops! I corrected my solution

Now it is correct