Existence of minimizers for nonconvex, noncoercive simple integrals.

We consider the problem of minimizing autonomous, simple integrals such as \min\,\left\{ \int_0^T f\left(x(t)\,,x^\prime(t)\right)\,dt\colon\,\, \text{$x\in AC{([0\,,T])}$, $x(0)=x_0$, $x(T)=x_T$} \right\}, \tag{$\cal{P}$} where $f:{\mathbb R}\times{\mathbb R} \to [0,\infty]$ is a possibly nonconvex function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ($\cal{P}^{\ast\ast}$)---obtained from ($\cal{P}$) by replacing f with its convex envelope f** with respect to the derivative variable $x^\prime$---admits a solution, we prove attainment for ($\cal{P}$) under mild regularity and growth assumptions on f and f**. We discuss various instances of growth conditions on f that yield solutions to the corresponding relaxed problem ($\cal{P}^{\ast\ast}$), and we present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp.