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Thread: Laplace equation

  1. #1
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    Laplace equation

    How can I find infinitely many functions $\displaystyle u:R^2 \rightarrow R$ that solve the Laplace equation? I think I need to find analytic functions $\displaystyle f$ in open subset $\displaystyle U \subset C$, but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
    $\displaystyle \frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0$
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  2. #2
    Super Member Rebesques's Avatar
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    You can find loads, say $\displaystyle x^2-y^2, \ {\rm e}^x{\rm cos}y, \ \rm log(x^2+y^2)^{1/2}$
    and try superposition.
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  3. #3
    MHF Contributor chiph588@'s Avatar
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    What about $\displaystyle \{ax^2-ay^2 | a \in \mathbb{R} \} $?
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  4. #4
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    Quote Originally Posted by jackie View Post
    How can I find infinitely many functions $\displaystyle u:R^2 \rightarrow R$ that solve the Laplace equation? I think I need to find analytic functions $\displaystyle f$ in open subset $\displaystyle U \subset C$, but I don't know how to use the Cauchy Riemann equations here to help me. Someone can help please.
    $\displaystyle \frac{\delta^2 u}{\delta x^2}+\frac{\delta^2 u}{\delta y^2}=0$
    Remember that every analytic function (in the open set $\displaystyle \Omega$ ) can be seen as $\displaystyle f= u +iv$ where $\displaystyle u,v: \Omega \longrightarrow \mathbb{R}$ are such that $\displaystyle u,v \in C^1(\Omega )$ and satisfy $\displaystyle \frac{ \partial u}{ \partial x} = \frac{ \partial v}{ \partial y}$ and $\displaystyle \frac{ \partial u}{ \partial y} = - \frac{ \partial v}{ \partial x}$ (with $\displaystyle u= u(x,y)$ and $\displaystyle v=v(x,y)$). Now assume that $\displaystyle u,v \in C^2 (\Omega)$ (if you haven't proved that the real and imaginary part of an anlytic function are of $\displaystyle C^{\infty}$ class) and using the C-R equations we get:

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

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

    And since $\displaystyle u,v \in C^2 (\Omega )$ we have that the mixed partial derivatives of $\displaystyle v$ are equal and so, adding the last two equations together we get that $\displaystyle u$ satisfies the Laplace equation in $\displaystyle \Omega$ (the same argument tells you that $\displaystyle v$ also satisfies Laplace in $\displaystyle \Omega$ ). 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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