Non-convex one-dimensional functionals with superlinear growth

We study the minimum problem for functionals of the form F(u) = integral(I) f(x,u(x), u'(x))dx, where the integrand f : I x R-d x R-d -> R is not convex in the last variable. We provide existence results in the Sobolev space W-1,W-1(I, R) analogous to the ones obtained in W-1,W-p (I, R-d) (p > 1) by a method inspired by integro-extremization and based on Euler equations. In addition, we treat functionals with nonsmooth Lagrangians and exhibit a comparison with a direct application of integro-extremality method to a class of functionals of sum type with a separate dependence on the components of the derivative u'.