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Maximum Modulus Theorem and Harmonic Functions

  1. Feb 17, 2009 #1
    1. The problem statement, all variables and given/known data

    Let u be harmonic on the bounded region A and continuous on cl(A). Then show that u takes its minimum only on bd(A) unless u is constant.

    2. Relevant equations

    incase you are used to diffrent notation, cl(a) is clousure bd(A) is boundary

    3. The attempt at a solution

    By the global maximum principle we have that (let m denote the minimum of u on bd(A))
    (i)u(x,y) >= m for (x,y) in A
    (ii) if u(x,y)=m for some (x,y) in A, then u is constant.

    Assume u is not constant.

    then we have that u(x,y)>m.

    I dont know where to go from here. I think i still need to show that the only possible place it can take its min is on bd(A). but i have no clue how to do that.
  2. jcsd
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