 Lavrentiev phenomenon

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 $I(u)={\int }_0^1(u^3-x{)}^2{u}^{\prime 6}dx$ 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, $|u^3-x|\ge 7x/8$. Holder inequality givesThus $I(u)\ge (49/4096)(3/5{)}^5/x_0\ge (49/4096)(3/5{)}^5>0$. 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.