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i try to show that the Dirichlet energy functional has a minimum subject to the constraint $\|u\|=1$.What do i have to do?

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Wouldn't u(x) = [1 1 1 ... 1] / sqrt(n) satisfy your conditions? The gradient is zero everywhere... –  pedrosorio Nov 4 '12 at 14:03
    
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Wouldn't u(x) = [1 1 1 ... 1] / sqrt(n) satisfy your conditions?

The gradient is zero everywhere and the Dirichlet energy is non-negative, therefore it must be a minimum.

Obviously this minimum is not unique as any constant function with norm 1 will satisfy your conditions.

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And how do i show hat it is a minimum? –  JamesBond Nov 4 '12 at 14:14
    
Thank you. What can we do if there is a boundary condition u=g? –  JamesBond Nov 4 '12 at 14:46
    
@JamesBond: The problem is essentially the same, just use a constant function $u$ whose norm is g. Consider accepting correct answers when you ask a question. –  pedrosorio Nov 4 '12 at 14:55
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