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**HA9 / HA9 Problem 1**

« **on:**March 24, 2015, 10:56:40 PM »

Find function $u$ harmonic in $\{x^2+y^2+z^2\le 1\}$ and coinciding with $g=z^4$ as $x^2+y^2+z^2=1$.

\begin{equation*}

u= z^4 + az^2(1-x^2-y^2-z^2)+b(1-x^2-y^2-z^2)^2

\end{equation*}

with unknown coefficients $a,b$.

It seems to me that we should have a harmonic polynomial of degree 4 if we want the function to coincide with $g(x)=z^4$ on ${x^2+y^2+z^2=1}$, I'm I missing something?

**Hint.**According to [Subsection 28.1] solution must be a harmonic polynomial of degree $4$ and it should depend only on $x^2+y^2+z^2$ and $z$ (Explain why). The only way to achive it (and still coincide with $g$ on $\{x^2+y^2+z^2=1\}$) is to find\begin{equation*}

u= z^4 + az^2(1-x^2-y^2-z^2)+b(1-x^2-y^2-z^2)^2

\end{equation*}

with unknown coefficients $a,b$.

It seems to me that we should have a harmonic polynomial of degree 4 if we want the function to coincide with $g(x)=z^4$ on ${x^2+y^2+z^2=1}$, I'm I missing something?