Maximum principle

  • #1
my function is u(x,y) = xy(x2 - y2 + 2).

Using the fact that this fxn is harmonic, how do I find the max of u on [0,1]x[0,1]?
 

Answers and Replies

  • #2
HallsofIvy
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Well, do you know the theorem that says if a function is harmonic on a set, then any extreme values must occur on the boundary of the set?

The boundary of [0,1]x[0,1] consists of 4 line segments but the function, reduced to those segments, is still harmonic so any extreme values must occur at the endpoints of those segments.
 
  • #3
so I guess the max would have to be at (1,1) then. So would i just let u(x,y) = u(1,1) and use the value that comes out?
 
  • #4
HallsofIvy
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Yes! The max and min must occur at the four "corners" (1,1), (1, 0), (0, 1) and (0, 0). You can see that u(1,1)= 2 while the others are all 0.
 
  • #5
Thank you so much. That was very helpful.
 

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