Power-law comparison for vanishing derivative regularization ($\min I_{\epsilon }=O({\epsilon }^{2/3})$)

For $I_{\epsilon }(u)={\int }_0^1[(u^4-x{)}^2+\epsilon |u^{\prime }{|}^{4/3}]dx$ 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 $(3\epsilon /(4\delta ))(1/4+\delta {)}^{4/3}$. Balancing ${\delta }^2$ and $\epsilon /\delta$ by $\delta ={\epsilon }^{1/3}$ gives an upper energy bound of order ${\epsilon }^{2/3}$. Nonnegativity then proves convergence of minimum energies to zero. This is a recovery construction; no claim of a matching lower asymptotic constant is needed.