HA 10, problem 3a

[Shaghayegh A]

link: http://www.math.toronto.edu/courses/apm346h1/20169/PDE-textbook/Chapter8/S8.P.html     (3a)

Since P(x,y,z) is a polynomial of degree 0, it is a constant. So $U=x^2+y^2+z^2-c_0 (x^2+y^2+z^2)$, but you can't write this as a sum of homogenous harmonic polynomials since there is a term c_0 remaining?

[Shentao YANG]

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

[Victor Ivrii]

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

[Shaghayegh A]

modified:
My solution is $U=x^2+y^2-z^2-\frac{1}{3} (x^2+y^2+z^2-1)$, 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 $x^2+y^2-z^2$

[Victor Ivrii]

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

[Shaghayegh A]

Oops! I corrected my solution

[Victor Ivrii]

Now it is correct