= Power-law comparison for vanishing derivative regularization
{title2=$\min I_\varepsilon=O(\varepsilon^{2/3})$}
For $I_\varepsilon(u)=\int_0^1[(u^4-x)^2+\varepsilon|u'|^{4/3}]dx$ with endpoint values zero and one, the zero-penalty minimizer $x^{1/4}$ has infinite <derivative> penalty. Use $u=x^{1/4+\delta}$ instead. The mismatch is at most $(32/27)\delta^2$, while the <derivative> penalty is $(3\varepsilon/(4\delta))(1/4+\delta)^{4/3}$. Balancing $\delta^2$ and $\varepsilon/\delta$ by $\delta=\varepsilon^{1/3}$ gives an upper energy bound of order $\varepsilon^{2/3}$. Nonnegativity then proves convergence of minimum energies to zero. This is a recovery construction; no claim of a matching lower asymptotic constant is needed.
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