= 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 $F(x,z,v)\geq c|v|^p-C$ 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 $F_\varepsilon=(z^4-x)^2+\varepsilon|v|^{4/3}$ is continuous, <convex> in $v$, and bounded below by $\varepsilon|v|^{4/3}$. The competitor $u(x)=x$ has finite energy. Thus for every $\varepsilon>0$ 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 \b[the same function $u_\varepsilon$ is a <global minimizer> in the full stated class].
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