Non-convex one-dimensional functionals with superlinear growth

Abstract

We study the minimum problem for functionals of the form \begin{equation*} \mathcal{F}(u) = \int_{I} f(x, u(x), u^\prime(x))\,dx, \end{equation*} where the integrand $f:I\times\mathbb R^d\times\mathbb R^d\to \mathbb R$ is not convex in the last variable. We provide existence results in the Sobolev space $W^{1,1}(I,\mathbb R)$ analogous to the ones obtained in $W^{1,p}(I,\mathbb R^ d)$ ($p>1$) 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 $u^\prime$.