Non-convex one-dimensional functionals with superlinear growth. (Q2128418)

The author studies the minimum problem for functionals \[F(u):=\int_I f(x,u(x),u'(x))\, dx\] where \(I\) is the interval \([a,b]\) and \(u\in W^{1,1}(I;\mathbb{R}^d)\) is subject to prescribe some boundary conditions. The function \(f\) is not convex with respect to the last variable and satisfies a superlinear growth at infinity. In a previous paper the author has studied the analogous problem in the space \(W^{1,p}(I;\mathbb{R}^d)\) with \(p>1\). The novelty of these two papers, the previous and the present one, lies in the fact that a new approach is used, inspired by the integro-extremality method which makes use of Euler-Lagrange equations. In this paper the same approach used before is used to treat the case \(p=1\), and moreover the cases of non-differentiable Lagrangians and of sum type Lagrangians (i.e. \(f(x,p,\xi)=h(x,p)+\sum_{i=1}^d g_i(x,\xi_i)\)). The sketched procedure is as follows: once considered \[ \bar{F}(u):=\int_I\bar{f}(x,u(x),u'(x))\, dx \] where \(\bar{f}\) is the lower convex envelope of \(f\) with respect to the last variable, the author assumes the existence of an index \(j\in\{ 1,\ldots ,d\}\) for which \begin{itemize}\item[\(i\, )\)] the derivative \(\bar{f}_{p_j}\) of \(\bar{f}\) with respect to the component \(p_j\) of \(p\) is non-negative;\item[\(ii\, )\)] for every point \((x,p,\xi)\) satisfying \(f(x,p,\xi)>\bar{f}(x,p,\xi)\) one has \(\bar{f}_{\xi_j}(x,p,\xi)=\alpha(x)\) with \(\alpha\) measurable;\item[\(iii\, )\)] \(\alpha\) is non-increasing.\end{itemize} These requirements are analogous to those assumed in the paper where the case \(p>1\) is treated. To prove existence of a minimizer for \(F\) the author considers a perturbed family of functionals, \[ F_{\epsilon}(u):=\int_I\big[\bar{f}(x,u(x),u'(x))+\epsilon\, \beta(u_j(x))\big]\, dx, \] where \(\epsilon\) is a positive parameter that will be sent to zero and \(\beta\) is a \(C^1\) strictly increasing and bounded function depending only on one variable and where \(j\) is the same index of \(i\, )\) and \(ii)\, \).