 Solution

ID: past-exam-of-the-partial-differential-equations-for-pure-and-applied-mathematicians-and-calculus-of-variations-course-of-the-university-of-oxford/2007/b/8/a/iii/solution

Write

l = in f_{A} I

. Since the limit is admissible, the assumed lower-semicontinuity inequality gives

l \leq I (u) \leq lim inf I (u_{j_{k}}) = l

. Thus $I (u) = l$ and $u$ is a global minimizer.

The extra integrand hypothesis in the Tonelli existence theorem for one dimension is convexity in the derivative variable for each fixed $(x, u)$ . Together with the stated continuity and coercive lower bound it gives the lower-semicontinuity property along the uniform-function/weak-derivative convergence just proved.

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