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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{, , } \right\}, \tag{} where is a possibly nonconvex function with either superlinear or slow growth at infinity. Assuming that the relaxed problem ()---obtained from () by replacing f with its convex envelope f** with respect to the derivative variable ---admits a solution, we prove attainment for () 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 (), and we present examples that show that the hypotheses on f and f** considered here for attainment are essentially sharp