We consider the functional F(v) = integral(a)(b) f(t,v'(t))dt in H-p = {v is an element of W-1,W-p : v(a) = 0, v(b) = d}, p is an element of [1, +infinity]. Under only the assumption that the integrand is Lcircle timesB(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.