On a class of non-convex non-coercive Bolza problems with constraints on the derivatives

We consider the minimization problem min{integral(a)(b) f (t, u'(t)) dt+l (u(a), u(b)); u. is an element of AC([a, b], R-n)}, where f: [a, b] x R-n --> Rboolean OR{+ infinity} is a normal integrand, l: R-n x R-n --> Rboolean OR {+infinity} is a lower semicontinuous function, and AC([a, b], R-n) denotes the space of absolutely continuous functions from [a, b] to R-n. We prove sufficient conditions for the existence of minimizers. We give applications to radially-symmetric variational problems, problems with unilateral constraints on the derivatives, the Newton problem of minimal resistance, models for Martensitic transformations, models in behavioral ecology, and the adiabatic model of the atmosphere.