Toronto Math Forum
APM3462015S => APM346Home Assignments => HA9 => Topic started by: Zacharie Leger 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 $x^2+y^2+z^2=1$.
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(1x^2y^2z^2)+b(1x^2y^2z^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?

Thanks, corrected.

This is Q1

Q1
From lecture section 28.1, we have
\begin{equation}
\Delta =\partial_\rho^2 + \frac{2}{\rho}\partial_\rho +
\frac{1}{\rho^2}\Lambda
\end{equation}
with
\begin{equation}
\Lambda:=
\bigl(\partial_{\phi}^2 + \cot(\phi)\partial_\phi \bigr) +\frac{1}{\sin^2(\phi)}\partial_{\theta}^2.
\end{equation}
Plug $u=P(\rho)Y(\phi,\theta)$ into $\Delta u=0$: \begin{equation}
P''(\rho)Y(\phi,\theta) + \frac{2}{\rho}P' (\rho)Y(\phi,\theta) + \frac{1}{\rho^2} P(\rho)\Lambda Y(\phi,\theta)=0
\end{equation}
which could be rewritten as
\begin{equation}
\frac{\rho^2 P''(\rho) + \rho P' (\rho)}{P(\rho)}+
\frac{\Lambda Y(\phi,\theta)}{Y(\phi,\theta)}=0
\end{equation}
and since the first term depends only on $\rho$ and the second only on$\phi, \theta$ we conclude that both are constant:
\begin{align}
\rho^2 P'' +2\rho P' = \lambda P,\\
\Lambda Y(\phi,\theta)=\lambda Y(\phi,\theta).
\end{align}
The first equation is of Euler type and it has solutions $P:=\rho^l$ iff
$\lambda= l(l+1)$. However if we are considering ball, solution must be infinitely smooth in its center due to some general properties of Laplace equation and this is possible iff $l=0,1,2,\ldots$ and in this case $u$ must be a polynomial of $(x,y,z)$.Which in this case, u depends on $x^2+y^2+z^2$ and $z$
Then suppose that
\begin{align}
u= &z^4 + a (1x^2y^2z^2) + bz^2 (1x^2y^2z^2) + c(1x^2y^2z^2) ^2\\\\
=&z^4 + a (1x^2y^2z^2) + b (z^2z^2x^2z^2y^2z^4) + c(1x^2y^2z^2) ^2
\end{align}
\begin{align}
\Delta u &= uxx+uyy+uzz\\
&=12 z^2 6a + 2b( 1 \rho^2 7 z^2) 6c +20c\rho^2\\\text{Suppose $1\rho^2z^2=0$}\\
&=12z^26a+2b(6z^2)6c+20c(1z^2)\\
&=12z^26a12bz^26c+20c20cz^2\\
&=(1212b20c)z^26a6c=0
\end{align}
Then we have
\begin{align}
1212b20c=0\\
6a6c=0
\end{align}
Then $a = c$, $b=1\frac{5}{3}c$, and c is arbitrary.

I found different values for a,b. :X

All do correct things but do errors in calculations. Actually, it was partially my fault as one needs to look at
\begin{align*}
u= &z^4 + a (1x^2y^2z^2) + bz^2 (1x^2y^2z^2) + c(1x^2y^2z^2) ^2\\\\
=&z^4 + a (1x^2y^2z^2) + b (z^2z^2x^2z^2y^2z^4) + c(1x^2y^2z^2) ^2
\end{align*}
Calculation of Laplacian applied to first three terms is very easy and we get
$$ 12 z^26a +b (22x^22y^216z^2) = 12 z^2 6a + 2b(1 \rho^2 7 z^2)$$
and for the fourth term $v=c(12\rho^2+\rho^4)$ we apply Laplacian in the spherical coordinates
$$v_{\rho\rho}+ 2\rho^{1}v_\rho= c(6 + 20\rho^2).$$ So
$$
\Delta u= 12 z^2 6a + 2b( 1 \rho^2 7 z^2) 6c +20c\rho^2
$$
Please, finish corrected version

\begin{equation}
\Delta =\partial_\rho^2 + \frac{2}{\rho}\partial_\rho +
\frac{1}{\rho^2}\Lambda
\end{equation}
with
\begin{equation}
\Lambda:=
\bigl(\partial_{\phi}^2 + \cot(\phi)\partial_\phi \bigr) +\frac{1}{\sin^2(\phi)}\partial_{\theta}^2.
\end{equation}
Plug $u=P(\rho)Y(\phi,\theta)$ into $\Delta u=0$: \begin{equation}
P''(\rho)Y(\phi,\theta) + \frac{2}{\rho}P' (\rho)Y(\phi,\theta) + \frac{1}{\rho^2} P(\rho)\Lambda Y(\phi,\theta)=0
\end{equation}
which could be rewritten as
\begin{equation}
\frac{\rho^2 P''(\rho) + \rho P' (\rho)}{P(\rho)}+
\frac{\Lambda Y(\phi,\theta)}{Y(\phi,\theta)}=0
\end{equation}
and since the first term depends only on $\rho$ and the second only on$\phi, \theta$ we conclude that both are constant:
\begin{align}
\rho^2 P'' +2\rho P' = \lambda P,\\
\Lambda Y(\phi,\theta)=\lambda Y(\phi,\theta).
\end{align}
The first equation is of Euler type and it has solutions $P:=\rho^l$ iff
$\lambda= l(l+1)$. However if we are considering ball, solution must be infinitely smooth in its center due to some general properties of Laplace equation and this is possible iff $l=0,1,2,\ldots$ and in this case $u$ must be a polynomial of $(x,y,z)$.Which in this case, u depends on $x^2+y^2+z^2$ and $z$
Then suppose that
\begin{align}
u= &z^4 + a (1x^2y^2z^2) + bz^2 (1x^2y^2z^2) + c(1x^2y^2z^2) ^2\\\\
=&z^4 + a (1x^2y^2z^2) + b (z^2z^2x^2z^2y^2z^4) + c(1x^2y^2z^2) ^2
\end{align}
\begin{align}
\Delta u &= u_{\xx}+u_{\yy}+u_{zz}\\
&=12 z^2 6a + 2b( 1 \rho^2 7 z^2) 6c +20c\rho^2\\\text{Suppose $1\rho^2z^2=0$}\\
&=12z^26a+2b(6z^2)6c+20c(1z^2)\\
&=12z^26a12bz^26c+20c20cz^2\\
&=(1212b20c)z^26a6c=0
\end{align}
Then we have
\begin{align}
1212b20c=0\\
6a6c=0
\end{align}
Then $a = c$, $b=1\frac{5}{3}c$, and c is arbitrary.
[/quote]

most part code is copied from Chen
\begin{equation}
\Delta =\partial_\rho^2 + \frac{2}{\rho}\partial_\rho +
\frac{1}{\rho^2}\Lambda
\end{equation}
with
\begin{equation}
\Lambda:=
\bigl(\partial_{\phi}^2 + \cot(\phi)\partial_\phi \bigr) +\frac{1}{\sin^2(\phi)}\partial_{\theta}^2.
\end{equation}
Plug $u=P(\rho)Y(\phi,\theta)$ into $\Delta u=0$: \begin{equation}
P''(\rho)Y(\phi,\theta) + \frac{2}{\rho}P' (\rho)Y(\phi,\theta) + \frac{1}{\rho^2} P(\rho)\Lambda Y(\phi,\theta)=0
\end{equation}
which could be rewritten as
\begin{equation}
\frac{\rho^2 P''(\rho) + \rho P' (\rho)}{P(\rho)}+
\frac{\Lambda Y(\phi,\theta)}{Y(\phi,\theta)}=0
\end{equation}
and since the first term depends only on $\rho$ and the second only on$\phi, \theta$ we conclude that both are constant:
\begin{align}
\rho^2 P'' +2\rho P' = \lambda P,\\
\Lambda Y(\phi,\theta)=\lambda Y(\phi,\theta).
\end{align}
The first equation is of Euler type and it has solutions $P:=\rho^l$ iff
$\lambda= l(l+1)$. However if we are considering ball, solution must be infinitely smooth in its center due to some general properties of Laplace equation and this is possible iff $l=0,1,2,\ldots$ and in this case $u$ must be a polynomial of $(x,y,z)$.Which in this case, u depends on $x^2+y^2+z^2$ and $z$
Then suppose that
\begin{align}
u= &z^4 + a (1x^2y^2z^2) + bz^2 (1x^2y^2z^2) + c(1x^2y^2z^2) ^2\\\\
=&z^4 + a (1x^2y^2z^2) + b (z^2z^2x^2z^2y^2z^4) + c(1x^2y^2z^2) ^2
\end{align}
\begin{align}
\Delta u &= u_{xx}+u_{yy}+u_{zz}\\
&=12 z^2 6a + 2b( 1 \rho^2 7 z^2) 6c +20c\rho^2\\\text{Suppose $1\rho^2z^2=0$}\\
&=12z^26a+2b(6z^2)6c+20c(1z^2)\\
&=12z^26a12bz^26c+20c20cz^2\\
&=(1212b20c)z^26a6c=0
\end{align}
Then we have
\begin{align}
1212b20c=0\\
6a6c=0
\end{align}
Then $a = c$, $b=1\frac{5}{3}c$, and c is arbitrary.
[/quote]

But equation must be fulfilled inside of domain! There will be no arbitrary constants!