Maximizing u(x,y): Using the Harmonic Function

[Somefantastik]

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]?

 

[HallsofIvy]

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.

 

[Somefantastik]

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?

 

[HallsofIvy]

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.

 

[Somefantastik]

Thank you so much. That was very helpful.