- #1
Mechmathian
- 35
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1. We look at a Laplace equation ( [tex]\Delta u(x,y) =o[/tex]) on a square [0, 1]* [0, 1]
If we know that [tex]u_{x=o}[/tex]= siny , [tex]u_{x=1}[/tex]= cosy
[tex]u_y|_{y=0}[/tex]= 0 , [tex]u_y|_{y=1}[/tex]= 0 we differentiate here by y. proove that |u|<=1.
We now know that the maximum of u has to be on the boundary. If it is greater then one, then it has to be on either y=0 or y =1.
If we know that [tex]u_{x=o}[/tex]= siny , [tex]u_{x=1}[/tex]= cosy
[tex]u_y|_{y=0}[/tex]= 0 , [tex]u_y|_{y=1}[/tex]= 0 we differentiate here by y. proove that |u|<=1.
The Attempt at a Solution
We now know that the maximum of u has to be on the boundary. If it is greater then one, then it has to be on either y=0 or y =1.