Past exam of the calculus of variations course of the University of Oxford / 2009 / 2 / b / iii

Table of contents
- Solution iii

Solution

Choose any nonzero admissible

u_{0}

and set

u = γ u_{0}

. Then

I_{λ} (γ u_{0}) = γ^{2} \int_{0}^{π} (u_{0}^{'2} - λ^{2} u_{0}^{2}) d x - \frac{γ ^{6}}{2009} \int_{0}^{π} u_{0}^{6} d x .

The coefficient of

- γ^{6}

is strictly positive, so the energy tends to minus infinity as

∣ γ ∣ \to \infty

. Hence

A in f I_{λ} = - \infty, 0 is not a global minimizer .

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