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

Superlinearity and continuity make $\Phi$ bounded below by a finite constant $M$. Consequently $I(u)\ge M(b-a)$ 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.