A Turnpike Result for a Class of Problems of the Calculus of Variations with Extended-Valued Integrands

We study the structure of approximate solutions of an autonomous variational problem with a lower semicontinuous integrand $$f:R^n\times R^n \to R^1\cup \{\infty\},$$ where $R^n$ is the $n$-dimensional Euclidean space. We are interested in a turnpike property of the approximate solutions which are independent of the length of the interval, for all sufficiently large intervals.