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Math Help - Laplace's equation on the unit square

  1. #1
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    Laplace's equation on the unit square

    Hello,

    I'm hoping for some help with the following question...

    v satisfies vxx + vyy = 0 on the open domain Q bounded by the unit square, with v(o,y)= y(1-y) and v = 0 on the other three sides of the square.
    Prove that, in the domain Q,

    0 < v(x,y) < 1/4(1-x).

    Any help would be much appreciated, my exam's in a couple of weeks and I'm stumped with this question!
    Thanks in advance
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  2. #2
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    Re: Laplace's equation on the unit square

    I think the left hand inequality is straightforward from the minimum principle. For the right hand side of the inequality, I migh suggest try letting

    w = v -  \dfrac{1}{4}(1-x)

    create new boundary conditions for w and use the maximum principle for w to show that w < 0 inside the square..
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  3. #3
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    Re: Laplace's equation on the unit square

    ahh, it seems so obvious now!! thank you so much
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