Laplace equation unit sphere

Laplec Equation is given by $\Delta u=0$ with $\Delta u=\sum_{i=1}^{n}u_{x_ix_i}$

Now I am confused with the following: What are solutions of Laplace Eq. for the inner of the unit sphere, which has on the surface the same values as $f_1(x_1,x_2,x_3)=x_1x_2, f_2(x_1,x_2,x_3)=x_1^2, f_3(x_1,x_2,x_3)=x_1^2+x_2^2, f_4(x_1,x_2,x_3)=x_1^2+x_2^2+x_3^2$

Do you have any suggestions?

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You should have at least checked that the given functions aren't harmonic themselves... $f_1$ is. Also, there's something very simple about the behaviour of $f_4$ on the sphere. –  user53153 Jan 16 '13 at 17:56
$f_1$ is the only function that satifies Laplace => Harmonic, right? $f_4$ describes a sphere, dou you mean that? The problem for me, I do not know how to calculate the solutions. –  Voyage Jan 16 '13 at 18:18

All of the given functions have constant Laplacian. For $f_1$ it's already zero. Others will need to be modified inside without changing their values on the boundary. Thus, the solutions will be $F_k=f_k+g_k$ where $\Delta g_k$ is an appropriate constant, and $g_k=0$ on the boundary. Hint below.
Hint: $g_k$ will be a constant multiple of $x_1^2+x_2^2+x_3^2-1$.
Thanks for your answer, but could you explain me a little more in detail how to receive $F_k$ ? –  Voyage Jan 16 '13 at 19:08
@voyage If function $f$ has $\Delta f = 5$ and another function $g$ has $\Delta g=2$, can you find a constant $c$ such that $f+cg$ is harmonic? –  user53153 Jan 16 '13 at 19:21