A GENERAL-APPROACH TO THE EXISTENCE OF MINIMIZERS OF ONE-DIMENSIONAL NONCOERCIVE INTEGRALS OF THE CALCULUS OF VARIATIONS

We present a general approach to get the existence of minimizers for a class of one-dimensional non-parametric integrals of the calculus of variations with non-coercive integrands. Motivated by the concrete applicative relevance of the problems, we extend the notion of solution to a class of locally absolutely continuous functions with generic boundary values. We extend by lower semi-continuity the functionals and we prove for them representation formulae. Assuming some structure conditions on the partial derivatives of the integrand, we obtain some W(loc)1, infinity a priori estimates that we use as a main tool to get existence. The results are then applied to get existence theorems for the classical Fermat's problem and for recent optimal foraging models of behavioural ecology.