i try to show that the Dirichlet energy functional has a minimum subject to the constraint $\|u\|=1$.What do i have to do?
Existence of minimizing function
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functional-analysis
calculus-of-variations
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0Wouldn't u(x) = [1 1 1 ... 1] / sqrt(n) satisfy your conditions? The gradient is zero everywhere... – 2012-11-04
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0You are spawning accounts like agent Smith in the second Matrix movie. **Please** consider registering your user. – 2012-11-04
1 Answers
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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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0Thank you. What can we do if there is a boundary condition u=g? – 2012-11-04
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0@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. – 2012-11-04
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0And how do i show hat it is a minimum? – 2012-11-04