= Lavrentiev phenomenon
{c}
{title2=$\inf_{W^{1,1}}I<\inf_{W^{1,\infty}}I$}
A Lavrentiev gap is a strict change in an <infimum> when the admissible regularity class is restricted, despite retaining the same endpoint values. For example $I(u)=\int_0^1(u^3-x)^2u'^6dx$ with $u(0)=0,u(1)=1$ has value zero at the absolutely continuous function $u=x^{1/3}$. Every Lipschitz competitor has a uniform positive lower energy bound. Indeed its ratio to $x^{1/3}$ starts at zero, so let $x_0$ be its first crossing of $u(x)=x^{1/3}/2$. Before that point, $|u^3-x|\geq7x/8$. <Holder inequality> gives
$$
\left(\frac{x_0^{1/3}}2\right)^6\leq\left(\int_0^{x_0}x^2|u'|^6dx\right)\left(\frac53x_0^{3/5}\right)^5.
$$
Thus $I(u)\geq(49/4096)(3/5)^5/x_0\geq(49/4096)(3/5)^5>0$. This proves a gap directly. The example's integrand lacks a uniform superlinear lower bound along its zero set; it illustrates the admissible-class issue rather than by itself furnishing a counterexample under stronger coercivity assumptions.
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