Variational problems with nonconvex, noncoercive, highly discontinuous integrands: characterization and existence of minimizers

We consider the functional F(v) = \int_a^b f(t,v′(t))dt in Hp = {v ∈ W^{1,p} : v(a) = 0, v(b) = d}, p ∈ [1, +∞]. Under only the assumption that the integrand is L ⊗ B_n-measurable, we prove characterizations of strong and weak minimizers both in terms of the minimizers of the relaxed functional and by means of the Euler–Lagrange inclusion. As an application, we provide necessary and sufficient conditions for the existence of the minimum, expressed in terms of a limitation on the width of the slope d.