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Math Help - Surface integral computation

  1. #1
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    Surface integral computation

    Okay, so this is a step in the proof to Liouville's theorem, and it comes from Fritz John's Partial Differential Equations 4th ed (p109, sec. 4.3). The problem is, it requires computation of a surface integral (I think), and I don't know how to do that.

    Here's the setup:

    Let u be harmonic in \mathbb{R}^n with \xi in the ball of radius a>0 about the origin. Let \omega_n=2\pi^{n/2}/\Gamma(n/2) be the surface area of the unit sphere S^{n-1}\subset\mathbb{R}^n (i.e. \omega_2=2\pi, \omega_3=4\pi, and so on.)

    We are also given the following identity:

    u_{\xi_i}(0)=\frac{n}{\omega_n a^{n+1}}\int_{|x|=a}x_i u(x)\;dS_x

    Frankly, I'm not sure what dS_x is supposed to denote. I assume it's the "surface integral" with respect to x, but then I'm not sure what the definition of a surface integral is, much less how to evaluate them routinely.

    Anyway, we have to use the above information to prove the following identity:

    |u_{\xi_i}(0)|\leq\frac{n}{a}\max_{|x|=a}|u(x)|

    Any ideas on how to do this? Any help would be much appreciated!
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  2. #2
    Super Member Rebesques's Avatar
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    Re: Surface integral computation

    I'm not sure what the definition of a surface integral is, much less how to evaluate them routinely.

    Maybe you should revisit your notes, it is quite helpful to be able to deal with them.





    Now for the estimate on the integral. Let \omega_n a^{n-1}=\int_{|x|=a}dS_x be the area of the a-radius n-sphere.
    Use the Cauchy-Schwartz inequality and basic properties of the integral to obtain

    |u_{\xi_i}(0)|\leq\frac{n}{\omega_n a^{n+1}}\left(\int_{|x|=a}\sum|x_i|^2dS_x\right)^{  1/2}\left(\int_{|x|=a}|u|^2dS_x\right)^{1/2}

    or

    |u_{\xi_i}(0)|\leq\frac{n}{\omega_n a^{n+1}}\left(a\left[\omega_n a^{n-1}\right]^{1/2}\right)\left((\sup|u|)\left[\omega_n a^{n-1}\right]^{1/2}\right)

    from where the result follows.
    Last edited by Rebesques; January 11th 2012 at 10:51 AM.
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