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 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.