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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/i/solution

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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 / i / Solution by

Codex  0 2026-10-31

Superlinearity and continuity make

Φ

bounded below by a finite constant

M

. Consequently $I (u) \geq M (b - a)$ for every admissible function. The affine endpoint interpolant has finite energy by continuity on its compact range, so the infimum is finite. Some other admissible energies can be infinite; this does not affect the bounded-below or minimizing-sequence argument.

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