f (x, y, p)

is continuous, convex in

p

, and has a uniform superlinear lower bound

f (x, y, p) \geq Θ (∣ p ∣) - C

with

Θ (r) / r \to \infty

, and at least one fixed-endpoint admissible absolutely continuous function has finite energy, then the energy attains its infimum in the fixed-endpoint

W^{1, 1}

class. One may take

Θ

nonnegative and convex. The lower bound makes derivatives uniformly integrable; endpoint control then gives uniform compactness of a minimizing sequence, and convexity supplies the required lower semicontinuity.

For the specified functional, the integrand is continuous, nonnegative, and convex in the derivative, with

f (x, y, p) \geq \frac{∣ p ∣ ^{52}}{52}, \frac{∣ p ∣ ^{52} /52}{∣ p ∣} ⟶ \infty.

u_{0} (x) = 2 x - 1

has the required endpoints and finite energy. Thus a global minimizer exists. In fact any finite-energy minimizer belongs to

W^{1, 52}

, since its derivative's 52nd power is integrable.

It is unique. The admissible class is convex, and the integrand is strictly convex in

(y, p)

∣ p ∣^{52}

and

∣ y ∣^{2011}

are strictly convex, with the latter multiplied by

e^{x} > 0

. If two distinct admissible finite-energy functions minimized the energy, their midpoint would have strictly smaller energy. Distinct continuous representatives differ on a set of positive measure, so the strict integral inequality really applies. This contradiction proves uniqueness.