Lavrentiev's phenomenon for totally unconstrained variational problems in one dimension (Q1596325)

The paper deals with the celebrated {Lavrentiev phenomenon} for variational integrals of the form \[ J(u)= \int^b_a f(x,u,u') dx. \] It is known that, even for smooth coercive functions \(f\), the infimum of \(J\) over a class \(Y\) of admissible functions may be strictly smaller than the infimum over another class \(X\) of more regular functions, with \(X\) dense in \(Y\). However, in all the examples available in the literature, the spaces \(X\) and \(Y\) satisfy the endpoints constraints \(u(a)= \alpha\) and \(u(b)= \beta\). In this paper the authors construct some examples, with nonnegative polynomial functions \(f\), which exhibit the Lavrentiev phenomenon even in the cases of one or both endpoints free. In the case of only one point constraint they prove the gap phenomenon for \[ f(x,s,z)= (s^5- x^3)^2 (z^{20}+ 1)^2 \] with \(Y= W^{1,1}(0,1)\) and \(X= W^{1,5/2}(0, 1)\), assuming the boundary condition \(u(0)= 0\). In the case of both free endpoints they prove the gap phenomenon for the same function \(f\) with \(Y= W^{1,1}(-1, 1)\) and \(X= W^{1,5/2}(-1, 1)\).