 Solution

A convenient form of the Tonelli existence theorem is: on a finite interval with prescribed endpoint values, if a continuous integrand $F(x,z,v)$ is convex in $v$ and satisfies $F(x,z,v)\ge c|v{|}^p-C$ for some $p>1$, then, provided a finite-energy admissible competitor exists, its integral attains a minimum in the corresponding $W^{1,p}$ class. The lower bound may more generally subtract an integrable function of $x$. Compact one-dimensional Sobolev embedding controls $z$, while convexity gives weak lower semicontinuity in the derivative.

Here $p=4/3$ and $F_{\epsilon }=(z^4-x{)}^2+\epsilon |v{|}^{4/3}$ is continuous, convex in $v$, and bounded below by $\epsilon |v{|}^{4/3}$. The competitor $u(x)=x$ has finite energy. Thus for every $\epsilon >0$ the Tonelli existence theorem gives an admissible minimizer in $W^{1,4/3}(0,1)$.

Any finite-energy member of the larger stated $W^{1,1}$ class necessarily has derivative in $L^{4/3}$ and bounded function values by absolute continuity; it therefore belongs to $W^{1,4/3}$. All other competitors have infinite energy. Hence the same function $u_{\epsilon }$ is a global minimizer in the full stated class.