Laplace equation

**jackie** · April 12th 2009, 10:30 PM

How can I find infinitely many functions $u:R^2 \rightarrow R$ that solve the Laplace equation? I think I need to find analytic functions $f$ in open subset $U \subset C$ , but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
$\frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0$

**Rebesques** · June 22nd 2009, 05:17 AM

You can find loads, say $x^2-y^2, \ {\rm e}^x{\rm cos}y, \ \rm log(x^2+y^2)^{1/2}$
and try superposition.

**chiph588@** · June 22nd 2009, 05:35 PM

What about $\{ax^2-ay^2 | a \in \mathbb{R} \}$ ?

**Jose27** · June 23rd 2009, 12:57 PM

Originally Posted by jackie

How can I find infinitely many functions $u:R^2 \rightarrow R$ that solve the Laplace equation? I think I need to find analytic functions $f$ in open subset $U \subset C$ , but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
$\frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0$

Remember that every analytic function (in the open set $\Omega$ ) can be seen as $f= u +iv$ where $u,v: \Omega \longrightarrow \mathbb{R}$ are such that $u,v \in C^1(\Omega )$ and satisfy $\frac{ \partial u}{ \partial x} = \frac{ \partial v}{ \partial y}$ and $\frac{ \partial u}{ \partial y} = - \frac{ \partial v}{ \partial x}$ (with $u= u(x,y)$ and $v=v(x,y)$ ). Now assume that $u,v \in C^2 (\Omega)$ (if you haven't proved that the real and imaginary part of an anlytic function are of $C^{\infty}$ class) and using the C-R equations we get:

$\frac{ \partial^2 u}{ \partial x^2} = \frac{ \partial^2 v}{ \partial x \partial y}$

$\frac{ \partial^2 u}{ \partial y^2} = - \frac{ \partial^2 v}{ \partial y \partial x}$

And since $u,v \in C^2 (\Omega )$ we have that the mixed partial derivatives of $v$ are equal and so, adding the last two equations together we get that $u$ satisfies the Laplace equation in $\Omega$ (the same argument tells you that $v$ also satisfies Laplace in $\Omega$ ). And so the real and imaginary part of an analytic function are solutions to the Laplace equation (ie. they're harmonic functions)

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