Past exam of the calculus of variations course of the University of Oxford / 2011 / 2 / a / i

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- Solution i

Solution

A weak local minimizer of a variational integral is an admissible

\overset{u}{ˉ}

for which there is

ε > 0

such that

I (v) \geq I (\overset{u}{ˉ}) whenever v \in A, ∥ v - \overset{u}{ˉ} ∥_{W^{1, \infty}} < ε .

Both the function and its first derivative must remain uniformly close, up to the almost-everywhere convention for the derivative.

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