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Mean Value Property in partial differential equations

  1. Mar 18, 2013 #1
    1. The problem statement, all variables and given/known data
    S is a ball of radius 1 in R^2;
    Δu=0 in S
    u=g in ∂S, g(x1,x2)>1 for any (x1,x2) in ∂S. Show that for any r satisfying 0<r<1 there is a point (x1,x2) in S such that u(x1, x2) >=1.

    2. Relevant equations

    using mean value formula: ∫u(y)dy=1/Vr^n(∫u(y)dy)

    3. The attempt at a solution

    g(x1,x2)=(1/2π)∫g dS, g(x1,x2)>1 on the boundary

    (1/2π)∫0 to 2π g dS=1/2π(2π*g^2/2), 2π cancels out, leaving g(x1,x2)=g^2/2>1
    u=g on the boundary so g(x1,x2)=u(x1,x2)>1.

    I dont think I did this right, I might have to use u(0,0), any hints will be appreciated
  2. jcsd
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